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+{"text":"\\section{Introduction}\n\nTheories of fully developed turbulence rely on the energy conservation in inviscid flows to derive the scaling laws of a turbulent system. However, energy is not the only quadratic invariant of the three dimensional (3D) incompressible Euler equation, there is also the helicity\n\\begin{equation}\nH =\\int{d^3 x \\, \\textbf{v}(\\textbf{x}) \\cdot \\textbf{w}(\\textbf{x})},\n\\end{equation}\nwhere $\\textbf{v}(\\textbf{x},t)$ is the velocity field and $\\textbf{w}(\\textbf{x},t)=\\nabla \\times \\textbf{v}$ is the vorticity field \\citep{Moffatt69}.\nHelical motions are observed in a wide variety of geophysical flows, such as supercell convective storms which seem to owe their long life and stability to the helical nature of their circulation \\citep{lilly86}. In astrophysics it is well known that helicity plays a key role in the generation of large scale magnetic fields through dynamo action \\citep{pouquet76, moffat78}. In hydrodynamic isotropic and homogeneous turbulence, recent studies of flows with net helicity confirmed that there is a joint cascade of energy an helicity to smaller scales \\citep{Borue, Chen03, Gomez}, as expected from theoretical arguments \\citep{Brissaud}. In the case of non-helical flows, velocity and vorticity tend to align locally creating patches of strong positive and negative helicities. In helical flows, helicity still fluctuates strongly around its mean value, both in space and time, creating locallized patches with helicity much larger or smaller than the mean. As helicity is not positive definite, the study of its turbulent fluctuations is difficult and has received less attention. The scaling laws followed by helical fluctuations are unclear, and few comparisons are available between helical and non-helical flows to understand their impact on the energy cascade. It has been proposed that helical structures may have an impact on the dynamics of turbulent flows as helical regions have the velocity and vorticity aligned, with a resulting quenching of the non-linear term in the Navier-Stokes equation \\citep{Tsinober}. Intermittency in the helicity flux has also been recently studied in numerical simulations \\citep{Chen203}, and the geometry of helical structures was considered using wavelet decompositions \\citep{Farge} and Minkowski functionals in the magnetohydrodynamic case \\citep{anvar07}.\n\nIn this work we use the the cancellation exponent \\citep{Ott92} to study the fast fluctuations of helicity in turbulent helical and non-helical flows. The cancellation exponent was introduced to study fast changes in sign of fields on arbitrarily small scales, and is based on the study of the inertial range scaling of a generalized partition function built on a signed measure. Under some conditions, it was shown to be related to the fractal dimension of structures in the flow and to the Hold\\\"er exponent \\citep{Sreenivasan}. It was used to characterize fluctuations in hydrodynamic turbulence and magnetohydrodynamic dynamos \\citep{Ott92}, as well as in two dimensional magnetohydrodynamic turbulence \\citep{Pouquet02, Jonathan}. In solar wind observations it was used to show that magnetic helicity is sign singular \\citep{Bruno97}. However, to the best of our knowledge, no studies of helical fluctuations in hydrodynamic turbulence have been done using the cancellation exponent. We analyze data stemming from direct numerical simulations (DNSs) of the Navier-Stokes equation in a three dimensional periodic box at large resolutions (up to $1024^3$ grid points) with different external mechanical forcings. Both helical and non-helical flows are considered. We find that helicity fluctuations are sign singular in both cases, and the scaling of helicity fluctuations seems to be independent of the net helicity of the flow. Furthermore, we obtain a relation between the helicity cancellation exponent, the fractal dimension of helical structures and the first order scaling exponent of helicity.\n\n\\section{The cancellation exponent}\nThe cancellation exponent \\citep{Ott92} was introduced to characterize the behavior of measures that take both positive and negative values, and to quantify a form of singularity where changes in sign occur on arbitrarily small scales. We can introduce the cancellation exponent for the helicity considering a hierarchy ${Q_{i}(l)}$ of disjoint subsets of size $l$ covering the entire domain occupied by the fluid $Q(L)$ of size $L$. For each scale $l$, a signed measure of helicity is introduced as\n\\begin{equation}\n\\mu_{i}(l)=\\int_{Q_{i}(l)}{d^3 x \\; H(\\textbf{x})}\\bigg\/\\int_{Q(L)}{d^3 x \\; |H(\\textbf{x})|},\n\\label{eq:mu}\n\\end{equation}\nwhere $H(\\textbf{x}) = \\textbf{v}(\\textbf{x}) \\cdot \\textbf{w}(\\textbf{x})$ is the helicity density such that the total helicity is $H=\\int d^3x H(\\textbf{x})$. Since the normalization factor in Eq. (\\ref{eq:mu}) is the integral of the absolute value of $H$ over the entire domain, the signed measure is bounded betwen $1$ and $-1$ and can be interpreted as the difference between two probability measures, one for the positive component and  another for the negative component of the helicity density. To study cancellations at a given length scale, we build the partition function\n\\begin{equation}\n\\chi(l)=\\sum_{Q_{i}(l)}{|\\mu_{i}(l)|}.\n\\label{eq:1}\n\\end{equation}\nIn the inertial range, the scaling law followed by the cancellations can be studied by fitting\n\\begin{equation}\n\\chi(l)\\sim l ^{-\\kappa},\n\\label{eq:chi}\n\\end{equation}\nwhere $\\kappa$ is the cancellation exponent. This exponent represents a quantitative measure of the cancellation efficiency. If such a scaling law exists and its range increases as the inertial range increases, the signed measure is called sign singular. In this case, changes in sign occur everywhere and in any scale considered in the limit of infinite Reynolds number. On the other hand, for a smooth field $\\kappa=0$.\n\n\\section{Code and numerical simulations}\nThe data we use for the analysis stems from DNS of the incompressible Navier-Stokes equation with constant mass density,\n\\begin{equation}\n\\frac{\\partial \\textbf{v}}{\\partial t}+\\textbf{v}\\cdot \\bigtriangledown \\textbf{v}=-\\bigtriangledown p+\\nu\\bigtriangledown^{2}\\textbf{v} + \\textbf{f},\n\\label{eq:Navier-Stokes}\n\\end{equation}\n\\begin{equation}\n\\bigtriangledown \\cdot \\textbf{v}=0.\n\\label{eq:Incompressibility}\n\\end{equation}\nwhere $\\textbf{v}$ is the velocity field, $p$ is the pressure (divided by the mass density), $\\nu$ is the kinematic viscosity, and $\\textbf{f}$ represents an external force that drives the turbulence. \n\nFor the analysis, we consider six numerical simulations at different Reynolds numbers and spatial resolutions, with different forcing functions that inject either energy or both energy and helicity. The simulations are listed in Table \\ref{tab_Tabla1} and described in more detail in a previous paper \\citep{Alexakis06}. Spatial resolutions range from $256^3$ to $1024^3$ grid points, with Reynolds numbers $R_e = UL\/\\nu$ (with $U$ the rms velocity and $L$ the integral scale) ranging form $\\approx 675$ to $\\approx 6200$, and Taylor Reynolds numbers $R_e = U\\lambda\/\\nu$ (where $\\lambda$ is the Taylor scale) ranging from $\\approx 300$ to $\\approx 1100$. Two forcing functions were considered: Taylor-Green (TG) forcing, which injects no helicity in the flow (although helicity fluctuations around zero develop as a result of nonlinearities in the Navier-Stokes equation), and Arn'old-Beltrami-Childress (ABC) forcing, which injects maximum (positive) helicity in the flow. The forcings were applied respectively in the shells in Fourier space with wavenumber $k_F=2$ and 3. All runs were continued for over ten turnover times after reaching the turbulent steady state. As a measure of the helicity content of each flow, we computed a relative helicity $\\rho_{H} = H\/(k_F E)$ (where $E$ is the total energy) averaged in time over the turbulent steady state of each run (see Table \\ref{tab_Tabla1}).\n\n\\begin{table}\n\\begin{center}\n\\begin{minipage}{7cm}\n\\begin{tabular}{@{}cccccccc@{}}\nRun & $N$  & $\\textbf{F}$ & $k_{F}$ & $\\rho_H$& $\\nu$ &$R_{e}$ & $R_{\\lambda} $    \\\\[3pt]\nT1 &$256$ & $TG$  & $2$ & $0.03$ & $2 \\times 10^{-3}$   &$675$  &  $300$ \\\\\nT2 &$512$ & $TG$  & $2$ & $0.01$ & $1.5 \\times 10^{-3}$ &$875$  &  $350$ \\\\\nT3 &$1024$& $TG$  & $2$ & $0.03$ & $3\\times 10^{-4}$    &$3950$ &  $800$ \\\\\nA1 &$256$ & $ABC$ & $3$ & $0.79$ & $2 \\times 10^{-3}$   &$820$  &  $360$ \\\\\nA2 &$512$ & $ABC$ & $3$ & $0.81$ & $2.5 \\times 10^{-4}$ &$2520$ &  $670$ \\\\\nA3 &$1024$& $ABC$ & $3$ & $0.83$ & $2.5 \\times 10^{-4}$ &$6200$ &  $1100$\n\\end{tabular}\n\\end{minipage}\n\\end{center}\n{\\caption[]{Runs used for the analysis. $N$ is the linear grid resolution, $\\textbf{F}$ is the forcing, either Taylor-Green (TG) or Arn'old-Beltrami-Childress (ABC), $k_{F}$ is the forcing wave number, $\\rho_H$ is the relative helicity, $\\nu$ is the kinematic viscosity, $R_{e}$ is the Reynolds number, and $R_{\\lambda}$ is the Reynolds number based on the Taylor scale.}\n\\label{tab_Tabla1}}\n\\end{table}\n\n\\section{Results and analysis}\nTo study scaling laws of helicity fluctuations in the inertial range of helical and non-helical flows, we performed a signed measure analysis and computed the cancellation exponent $\\kappa$ for all runs. Following Eq. (\\ref{eq:chi}), this can be done by computing $\\chi(l)$ and fitting $\\chi(l) \\sim l^{-\\kappa}$ in the inertial range.\n\nAlthough a detailed analysis of the simulations can be found in \\citet{Alexakis06}, in Fig. \\ref{fig:espectros} we show as a reference the energy spectrum of the two simulations at the larger Reynolds numbers attained (runs A3 and T3). The spectra present a short inertial range followed by a bottleneck, as is usual in simulations of hydrodynamic turbulence. The runs also have a range with approximately constant energy flux, as well as a range of scales consistent with Kolmogorov 4\/5 law when third-order structure functions are computed. As usual, these ranges have slightly different extensions depending on the quantity studied, and it will be shown that the cancellation exponent shows a scaling range comprised betwen $l\\approx 0.08$ and $\\approx 0.7$, which correspond roughly to wavenumbers $k\\approx 8$ and $\\approx 80$, which is wider than but includes the inertial range of both runs. At the lower resolution, the inertial range in the energy spectrum and flux, or in the third-order structure function, is shorter and harder to identify.\n\\begin {figure}\n\\begin{center}\n\\includegraphics[width=14.0cm]{fig1.pdf}\n\\caption {Energy spectrum for the A3 (left) and T3 (right) runs. Kolmogorov scaling is indicated as a reference.}\n\\end{center}\n\\label{fig:espectros}\n\\end{figure}\n\nFig. \\ref{fig:cancelacionp} (left) shows the signed measure of helicity as a function of the scale for runs T1, T2, and T3 from top to bottom (non-helical, with increasing Reynolds numbers). As the Reynolds number increases, a wider scaling range can be identified. The values obtained for the cancellation exponent are $\\kappa = 0.92 \\pm 0.09$, $0.75 \\pm 0.03$, and $0.73\\pm0.01$ respectively for runs T1, T2, and T3. The values obtained and the dependence with Reynolds number support the idea that in the limit of vanishing viscosity, helicity fluctuations are sign singular, i.e., that fast changes in sign of helicity occur everywhere in arbitrarily small scales in the flow. This result is consistent with helicity cascading towards small scales, and with observations of the helicity distribution being highly intermittent \\citep{Chen203} (albeit the previous studies are only for flows with net helicity).\n\\begin {figure}\n\\begin{center}\n\\includegraphics[width=6.5cm]{fig2a.pdf}\n\\includegraphics[width=6.5cm]{fig2b.pdf}\n\\end{center}\n\\caption {Signed measure of helicity as a function of the scale for runs with increasing resolution, from top to bottom: TG forcing (left), and ABC forcing (right) after subtracting the mean value of helicity.  Slopes with the cancellation exponent are given as a reference.}\n\\label{fig:cancelacionp}\n\\end{figure}\n\nAs mentioned earlier, the flows with ABC forcing are helical. For such flows the direct cascade of helicity and its intermittency has been studied before using spectral quantities. Although helicity fluctuations develop in these flows, given the predominant sign of helicity, fluctuations occur around the mean value and less changes in sign take place. Indeed, if the signed measure of helicity and the cancellation exponent is computed for run A3, we obtain $\\kappa = 0.26 \\pm 0.04$. This smaller value of the exponent is consistent with the less cancellations, and the larger error is associated to the fact that larger fluctuations are observed in the signed measure as less events whith changes of sign are available. However, we are interested in helicity fluctuations around the mean value, and to correctly consider these fluctuations the mean value of helicity $H$ was subtracted from the helicity density in runs A1, A2, and A3 prior to the analysis. The resulting signed measures are also shown in Fig. \\ref{fig:cancelacionp} (right). In this case we also observe scaling of $\\chi(l)$ expanding over a wider range of scales as the Reynolds number increases, indicating that also for helical flows helicity fluctuations are sign singular.\n\nThe results obtained for the helicity cancellations in helical and non-helical runs are similar. The cancellation exponents are $\\kappa = 0.89 \\pm 0.07$, $0.80 \\pm 0.03$, and $0.72\\pm0.01$ respectively for runs A1, A2, and A3. At the highest resolution three ranges can be identified in both the A3 and T3 runs. At the smaller scales, viscous effects dominate and the flow is smooth. This results in less changes of sign in helicity and a shallower distribution of the signed measure. The slope tends to zero, as expected for a smooth flow, and the signed measure tends to one for the smallest resolved scale. At intermediate scales an inertial range is observed, and at larger scales the results are affected by the external forcing. In this scales, large fluctuations in the value of $\\chi(l)$ are observed as a result. As mentioned before, the inertial range of $\\chi(l)$ is wider than the inertial range observed in the energy spectrum. This is consistent with observations of the helicity spectrum decaying slower than the energy spectrum near the dissipative range in helical turbulence \\citep{Alexakis06}. \n\nThe cancellation exponent obtained in all runs as a function of the Taylor based Reynolds number is shown in Fig. \\ref{fig:kappa} (a). For both the helical and non-helical runs $\\kappa$ decreases with the Reynolds number, and for the largest Reynolds numbers studied the value of $\\kappa$ seems to saturate and is the same within error bars for both helical and non-helical flows. Note that because of the different forcing functions, the helical and non-helical runs have slightly different Reynolds numbers even when the same grid resolution is used. The similarities between the scaling laws of the helical and non-helical flows suggests that statistics of the helicity fluctuations are the same in both cases. This is further confirmed by a histogram of the helicity fluctuations around its mean value for the T3 and A3 runs. As Fig. \\ref{fig:kappa} (b) shows, turbulent fluctuations of helicity are similar, with small differences on the tails. \n\n\\begin {figure}\n\\begin{center}\n\\includegraphics[width=6.5cm]{fig3a.pdf}\n\\includegraphics[width=6.5cm]{fig3b.pdf}\n\\end{center}\n\\caption {(a) Cancellation exponent $\\kappa$ as a function of the Taylor based Reynolds numbers for the helical (triangles) and the non-helical (diamonds). (b) Histogram of helicity fluctuations for the T3 (solid) and A3 (dashed) runs.}\n\\label{fig:kappa}.\n\\end{figure}\n\nHelicity fluctuations are sign singular. The fast oscillations in sign point to a highly intermittent quantity that fluctuates rapidly in localized structures at arbitrarily small scales for infinite Reynolds numbers. As mentioned in the introduction, the cancellation exponent can be related to the geometry of such structures, and under some conditions these relations can be rigorously derived \\citep{Sreenivasan}. The relations can be found also using simple geometrical arguments. In the following, we extend the analysis of \\citet{Pouquet02} for the current density in magnetohydrodynamics to consider the case of kinetic helicity in hydrodynamic turbulence. We assume the helicity is spatially correlated (although not necessarily locally smoth) in $D$ dimensions and uncorrelated in $3-D$ dimensions, where $3$ follows from the dimension of the space. With this choice, a correlated helicity distribution has $D=3$, and a completely uncorrelated helicity distribution has $D=0$. For intermediate cases, we can estimate the signed measure of helicity as follows\n\\begin{eqnarray}\n\\chi(l) = \\sum_{Q_{i}(l)}{\\left|\\int_{Q_i(l)}{d^3 x \\, H({\\bf x})} \\, \\bigg{\/}\\nonumber \n    \\int_{Q(L)}{d^3 x \\, |H({\\bf x})|}\\right|}\\\\  \n           \\sim \\frac{1}{L^3\\langle H^{2}\\rangle^{1\/2}}\n            \\left(\\frac{l}{L} \\right)^3\\left|\\int_{Q(l)}{d^3x H(\\textbf{x})}\\right|, \n\\label{eq:fractal}\n\\end{eqnarray}\nwhere we used homogeneity to replace the sum over all subsets $Q_{i}(l)$ by $(L\/l)^3$ (the total number of terms in the sum) times the integral over a generic box $Q(l)$ of size $l$. Since the absolute values in the denominator prevent any cancellation, the integral below was approximated by the characteristic value $L^{3}H_{rms}$ where $H_{rms}=\\langle H^{2}\\rangle^{1\/2}$ is the rms value of the helicity density.\n\nIn the inertial range, and still under homogeneity asumptions, the helicity follows some scaling law which can be represented by helicity structure functions $\\left\\langle \\delta H(s)\\right\\rangle \\sim s^{h}$, where $s=|\\textbf{s}|$ represents a lineal increment. We can then split the integral over the domain $Q(l)$ in integrals over subdomains with volume $\\lambda^3$ such that they separate the contributions from domains where the helicity is correlated of domains where the helicity is uncorrelated in space. Then, the remaining integral in Eq. (\\ref{eq:fractal}) can be estimated as \\citep{Pouquet02}\n\\begin{equation}\n\\left|\\int_{Q(l)}{d^3x \\; H(\\textbf{x})}\\right| \\sim\n\\langle H^{2}\\rangle^{1\/2} \\int_{Q(l)} d^Ds \\left( \\frac{s}{\\lambda}\\right)^{h}\\int_{Q(l)} d^{3-D}s .\n\\label{eq:contri}\n\\end{equation}\nThe uncorrelated dimensions give a value proportional to the square root of their volume $(l\/ \\lambda)^{(3-D)\/2}$. The correlated dimensions give a contribution proportional to $(l\/\\lambda)^{h+D}$, wich for $h=0$ (locally smoth helicity) is proporcional to their volume. Replacing this results in Eq. (\\ref{eq:fractal}), we get that the signed measure of helicity $\\chi(l)$ scales as\n\\begin{equation}\n\\chi(l) \\sim l^{-\\left(\\frac{3-D}{2}-h\\right)}. \n\\label{eq:final}\n\\end{equation}\n\n\\begin {figure}\n\\begin{center}\n\\includegraphics[width=14.0cm]{fig4.pdf}\n\\end{center}\n\\caption {Contour levels of helicity density in a slice of the T3 run. Only $1\/4$ of the slice is shown. Left: contours with $H({\\textbf x})=0$, i.e., places where cancellations take place. Right: contours with $H({\\textbf x})=12$. Note how contour levels of zero helicity fill the space, while white patches are observed in the contour levels of non-zero helicity, as well as filamentary structures.}\n\\label{fig:filamentos}\n\\end{figure}\n\nThus the cancellation exponent $\\kappa$, the fractal dimension $D$, and the scaling exponent $h$ are related through\n\\begin{equation}\n\\kappa=\\frac{3-D}{2}-h.\n\\label{eq:rel}\n\\end{equation}\nA completely smooth field ($D=3$) has $\\kappa=h=1$, in agreement with the definition of $\\kappa$ and of the first order (H\\\"older) scaling exponent $h$ for the field (see \\citet{Bruno97}). In a turbulent flow, the global helicity is observed to follow Kolmogorov scaling, and therefore we can assume $h=1\/3$ \\citep{kurien, Alexakis06} . Then from the value of $\\kappa=0.73\\pm0.01$ it follows that helical structures are filamentary with fractal dimension $D=0.873\\pm0.005$, in good agreement with previous observations of helical vortex filaments \\citep{Tsinober, Farge, Chen203, Alexakis06}. As an illustration, Fig.\\ref{fig:filamentos} shows a slice of the helicity density in the T3 run. Filamentary structures can be identified, specially for the contour levels corresponding to large concentration of helicity. Structures less elongated were confirmed to correspond to filaments that cross the slice perpendicularly. \n\n\\section{Conclusions}\nSigned measures of helicity were calculated for turbulent flows with different Reynolds numbers and helicity contents. For both helical and non-helical flows cancellation exponents were calculated, obtaining a value compatible with $\\kappa=0.73\\pm0.01$ for helicity fluctuations for both cases at the largest Reynolds number considered. The scaling range increases with the Reynolds number, and the value of $\\kappa$ obtained indicates helicity is sign singular in the limit of infinite Reynolds number. Simple geometrical arguments were used to link the cancellation exponent $\\kappa$ to the fractal dimension $D$ of the structures. To do this we assumed a single scaling exponent $h$ for the lineal helical increments. Assuming $h=1\/3$, a value consistent with numerical results, we obtain that helicity fluctuations are filamentary with fractal dimension of $D=0.873\\pm0.005$, also in agreement with observations.\n\n\\begin{acknowledgments}\nThe authors acknowledge support from Grants No. UBACYT X468\/08 and PICT-2007-02211. PDM acknowledges support from the Carrera del Investigador Cient\\'{\\i}fico of CONICET.\n\\end{acknowledgments}\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThis paper continues the theory of dispersive blow up which was \ninitiated and developed  in \\cite{BS1} and \\cite{BS2}.  The present contribution is especially relevant to nonlinear \nSchr\\\"odinger-type equations, and includes theory for the Davey-Stewartson and the Gross-Pitaevskii equations. The work on \n Schr\\\"odinger equations substantially extends the results already available in \\cite{BS2}.  \n\nDispersive blow up of wave equations is a phenomenon of focusing of \nsmooth initial disturbances with finite-mass (or, finite-energy, depending on the physical context) that \nrelies upon the dispersion relation guaranteeing that, in the\nlinear regime, different\nwavelengths propagate at different speeds.   This is especially the \ncase for models wherein the linear dispersion is unbounded, so that energy can\nbe moved around at arbitrarily high speeds, but even bounded dispersion\ncan exhibit this type of singularity formation.  \n\nTo be more concrete, consider the Cauchy problem \nfor the linear (free) Schr\\\"odinger equation\n\\begin{equation}\\label{eq:linSch}\ni \\partial_t u + \\Delta u = 0, \\quad u\\big|_{t=0} = u_0(x),\n\\end{equation}\nwhere $x\\in {\\mathbb R}^n$ for some $n\\in {\\mathbb N}$.  For $u_0\\in L^2({\\mathbb R}^n)$, elementary \nFourier analysis shows the solution to this initial-value problem is\n\\begin{equation}\\label{eq:formula}\nu(x,t) = e^{i t \\Delta} u_0(x):= \\frac{1}{(2\\pi)^n} \\int_{{\\mathbb R}^n} e^{-i t |\\xi|^2} \\widehat u_0(\\xi) e^{i \\xi\\cdot x} d\\xi.\n\\end{equation}\nHere, $\\widehat u_0$ denotes the Fourier transformed initial data, {\\it viz}.  \n\\[\n\\mathcal Fu_0(\\xi) \\equiv\\widehat u_0(\\xi)= \\int_{{\\mathbb R}^n} u_0(x) e^{-i \\xi \\cdot x} \\, dx.\n\\]\nThe corresponding inverse Fourier transform will be denoted by $\\mathcal F^{-1}$. From \\eqref{eq:formula}, it is immediately inferred that for any $s \\in {\\mathbb R}$, solutions lie in $C({\\mathbb R};H^s)$ \nwhenever $u_0$ lies in the $L^2$-based Sobolev space $H^s$.  \nMoreover,  the evolution  preserves all these Sobolev-space   \nnorms, which is to say \n\\[\\| u(\\cdot, t) \\|_{H^s({\\mathbb R}^n)} = \\| u_0 \\|_{H^s(R^n)} \\] \nfor $t\\in {\\mathbb R}$.   In certain applications of this model, the case $s=0$ in the last formula\n corresponds to \nconservation of total mass in the underlying physical system.\n\nHowever, in Theorem 2.1 of \\cite{BS2}, it was shown that for any given  point $(x_*, t_*)\\in  {\\mathbb R}^{n}\\times {\\mathbb R}_+$, \nthere exists initial data  $u_0\\in C^\\infty({\\mathbb R}^n)\\cap L^2({\\mathbb R}^n)\\cap L^\\infty({\\mathbb R}^n)$ \nsuch that the solution $u(x,t)$ of the corresponding initial-value problem \\eqref{eq:linSch} \nfor the free Schr\\\"odinger equation  is continuous on ${\\mathbb R}^n\\times {\\mathbb R}_+ \\setminus \\{(x_*, t_*)\\}$, but\n\\[\n\\lim _{(x,t)\\in {\\mathbb R}^{n}\\times {\\mathbb R}_+\\to (x_*, t_*)} | u(x,t)| = + \\infty.\n\\] \nThis fact is referred to as (finite-time) dispersive blow up and will \nsometimes be abbreviated  DBU in the following. \nThe analogous phenomena also appears in other linear dispersive equations, such \nas the linear  Korteweg-de Vries equation \\cite{BBM} and the  free surface \nwater waves system \nlinearized around the rest  state \\cite{BS2}.\n\n\nAt first sight, one would expect that nonlinear terms would destroy \ndispersive blow up.  What is a little surprising is that even the \ninclusion of physically relevant nonlinearities in various models \nof wave propagation does not prevent dispersive blow up.  Indeed, \ntheory shows in some important cases that initial data \nleading  to this focusing singularity under the linear evolution continues to \nblow up in exactly the same way when nonlinear terms are \nincluded.  \nIn \\cite{BS1}, this was shown to be \ntrue for the Korteweg-de Vries equation, a model for shallow water waves and other \nsimple wave phenomena.  This result and\nanalogous dispersive blow up theory in \\cite{BS2} for  \nsolutions of the one-dimensional nonlinear Schr\\\"odinger equations,\n\\begin{equation}\n\\label{**}\ni \\partial_t u + \\partial^2_{x} u \\pm |u|^pu=0, \\quad u\\big|_{t=0} = u_0(x),\n\\end{equation}\nwhere $x\\in {\\mathbb R}$ and $p \\in (0,3)$,\n lead to the \nspeculation that such focusing might be one road to the formation of rogue waves \nin shallow and deep water and in nonlinear optics   \n(see \\cite{D, DGE, KPS, SRKJ}).  \n\nThe analysis put forward in \\cite{BS1} and \\cite{BS2} revolves around providing bounds on\nthe nonlinear terms in a Duhamel representation of the \nevolution. Because the phenomenon is due to the linear terms in the equation,\ndata of arbitrarily small size will still exhibit dispersive blow up, and\nindeed it can be organized to happen arbitrarily quickly.   This \nemphasizes the linear aspect of these singularities and differentiates\nit from the  blow up that occurs for some of the same\nmodels when the nonlinear term is focusing and sufficiently strong (see \\cite{SuSu} for a general overview\n of this aspect of Schr\\\"odinger equations). Moreover, even though the theory begins by showing that there are specific initial data that lead to \ndispersive blow up, the result is in fact self-improving.  Dispersive blow up\n continues to hold if this special initial data is subjected to a smooth perturbation. \n The theory further implies that there is $C^\\infty$ initial data with compact \nsupport which can be taken as small as we like that will, in finite time, become large \nin a neighborhood of a prescribed spatial point (see Remark 3.5 for more details). \nDispersive blow up thereby also serves to demonstrate ill-posedness of the considered models in $L^\\infty$--spaces.  \n\n\n\nThe aim of the present work is to generalize the results mentioned above in several respects. \nMost importantly, the dispersive blow up that in \\cite{BS2} was obtained for \\eqref{**}  will be shown to\nhold true of nonlinear Schr\\\"odinger \nequations in {\\it all dimensions} $n \\geq 1$ and for \nthe whole range of nonlinearities $p\\geq \\lfloor\\frac{n}{2}\\rfloor$, with or without a (possibly unbounded) real-valued\npotential.  Here and in the following, for $\\mu \\in {\\mathbb R}$, the quantity $ \\lfloor\\mu \\rfloor$ is\n the greatest integer less than or equal\nto $\\mu$.  Higher-order Schr\\\"odinger equations are also countenanced.  Our theory relies \nespecially on the results of Cazenave and Weissler established in \\cite{CW}. \nIn addition to  Schr\\\"odinger equations, dispersive \n blow up is proved for  Gross-Pitaevskii equations \n with non-trivial boundary conditions at infinity and for the Davey-Stewartson systems.\n\nAs a by-product of our analysis, a sharp global smoothing effect is obtained \nfor the nonlinear integral term in \nthe  equation derived from \\eqref{**} by use of Duhamel's formula.\n\nThe paper proceeds as follows:   Section 2 is concerned with some preliminaries \nwhich are mostly linear in nature.  Dispersive blow up for nonlinear Schr\\\"odinger \nequations is tackled in Section 3. Section 4 deals with the sharp \nglobal smoothing property mentioned earlier.  This latter theory, in addition to being \nof interest in itself, is used to complete the analysis in Section 3.   \nDispersive blow up for the Davey-Stewartson \nsystems then follows more or less as a corollary to the results in Sections 2 and 3.\n  The Gross-Pitaevskii equation takes center stage in \nSection 5 whilst higher-order Schr\\\"odinger equations are studied in Section 6.  \n\n\n\n\n\n\\section{Mathematical preliminaries} \\label{sec:prelim}\n\nIn this section, a review of the basic idea behind dispersive blow up \nis provided in the context of nonlinear\nSchr\\\"odinger equations.  Parts of the currently available\n theory for the linear Schr\\\"odinger group \nare also recalled in preparation for the analysis in Section 3.   \n\n\n\\subsection{Dispersive blow up in linear Schr\\\"odinger equations} \\label{sec:lin}\n\nTo understand the appearance of dispersive blow up in the solution of \\eqref{eq:linSch},\nstart by explicitly computing the inverse Fourier transformation in \\eqref{eq:formula} to\nsee that the free Schr\\\"odinger group admits the \nrepresentation \n\\begin{equation}\\label{eq:group}\nu(x,t)  = \\frac{1}{(4i \\pi t)^{n\/2}} \\int_{{\\mathbb R}^n} e^{i\\frac{|x-y|^2}{4t}} u_0(y)\\, dy,\\quad \\text{for $t\\not =0.$}\n\\end{equation}\nThis representation formula is the starting point of the following lemma.\n\\begin{lemma}\\label{lem:IC}\nLet $\\alpha \\in {\\mathbb R}$, $q\\in {\\mathbb R}^n$ and \n\\begin{equation*} \nu_0(x) = \\frac{e^{- i  \\alpha |x - q|^2}}{(1+|x|^2)^m}\\ ,\\quad \\text{with $\\frac{n}{4} < m \\le \\frac{n}{2}$.}\n\\end{equation*}\nThen, $u_0 \\in C^\\infty({\\mathbb R}^n)\\cap L^2({\\mathbb R}^n)\\cap L^\\infty({\\mathbb R}^n)$ and the associated global in-time solution \n$u \\in C({\\mathbb R}; L^2({\\mathbb R}^n))$ of \\eqref{eq:linSch} has the following properties.\n\\begin{enumerate}\n\\item At the point $(x_*,t_*)=(q,\\frac{1}{4\\alpha})$, the solution $u$ in \\eqref{eq:group} blows up, \nwhich is to say, \n\\[\n\\lim _{(x,t)\\in {\\mathbb R}^{n+1}\\to (x_*,t_*)} | u(x,t)| = + \\infty,\n\\]\n\\item it is a continuous function of $(x,t)$ on  $\\, {\\mathbb R}^n\\times {\\mathbb R} \\setminus \\{t_*\\}$ and\n\\item  $u(x,t_0)$ is a continuous function of $x \\in {\\mathbb R}^n \\setminus \\{x_*\\}$.\n\\end{enumerate} \n\\end{lemma}\n\\begin{proof}   First note that $u_0 \\in C^\\infty({\\mathbb R}^n)\\cap L^2({\\mathbb R}^n)\\cap L^\\infty({\\mathbb R}^n)$, that \n$u \\in C({\\mathbb R}; L^2({\\mathbb R}^n))$ and that the $L^2$--norm of $u$ is constant in view of mass conservation. On the other hand, \n evaluating \\eqref{eq:group} at $t=\\frac{1}{4\\alpha}$ for this particular $u_0$   gives\n\\[\nu\\left(x, \\frac{1}{4\\alpha}\\right)=\\left( \\frac{\\alpha}{i\\pi} \\right)^{n\/2} e^{i \\alpha (|x|^2-|q|^2)} \\int_{{\\mathbb R}^n} e^{-2i \\alpha y\\cdot (x-q)} \\frac{dy}{(1+|y|^2)^m}.\n\\]\nThus at $x=q$, it transpires that \n\\[\n\\left| u\\left(q, \\frac{1}{4\\alpha}\\right) \\right | =  \\left( \\frac{\\alpha}{\\pi} \\right)^{n\/2}   \\int_{{\\mathbb R}^n}  \\frac{dy}{(1+|y|^2)^m}  = +\\infty\n\\]\nprovided $m \\le \\frac{n}{2}$.  Assertions (ii) and (iii) can then be proved by the same arguments as in the proof of \\cite[Theorem 2.1]{BS2}. \nIn this endeavor, it is useful to note that $(1+x^2)^{-m}$ is closely related to the inverse Fourier transform of the modified Bessel functions $K_\\nu(|x|)$, where \n$\\nu = \\frac{n}{2}-m$. \n\\end{proof}\n\nIn other words,  for any given $q\\in{\\mathbb R}^n, \\alpha \\in {\\mathbb R}$, we have\nconstructed an explicit family of bounded smooth initial data (with finite mass) for which the solution of \nthe free Schr\\\"odinger equation \\eqref{eq:linSch} exhibits dispersive blow up at  the point $(x_*, t_*) = (q, \\frac{1}{4\\alpha})$ in space and time.   This result can be immediately generalized in \nvarious ways.  The  following  sequence of remarks indicates some of them.  \n\n\n\\begin{remark}\\label{amp}\nThe same argument shows that any initial data of the form\n\\[\nu_0(x) = e^{- i  \\alpha |x - q|^2} a(x),\n\\]\nwith an amplitude $a\\in C^\\infty({\\mathbb R}^n)\\cap L^2({\\mathbb R}^n)\\cap L^\\infty({\\mathbb R}^n)$ but $a\\not \\in L^1({\\mathbb R}^n)$ \nwill exhibit   dispersive blow up. \nUsing the superposition principle,  one can construct initial data which yield \ndispersive blow up at any\ncountably many isolated points in space-time ${\\mathbb R}^n \\times {\\mathbb R}$. In addition, multiplying $u_0$ by $\\delta$\nwith  $0<\\delta \\ll1$, allows for initial \ndata which are arbitrarily small, but which nevertheless blow up at $(x_*, t_*) $. \nBy  a suitable spatial truncation of $u_0$, one can also construct small, \nsmooth, bounded initial data with finite mass,  all of whose derivatives also have finite \n$L^2$--norm,  such that the corresponding solution $u$ remains smooth \nbut achieves arbitrarily large values at a given  point in space-time (see \\cite{BS2} for more details). \n\\end{remark}\n\n\n\n\n\n\\begin{remark}\\label{frac}\nIt was also proven in \\cite{BS2} that dispersive blow up  holds true for the  class \n\\begin{equation*}\\label{fracS}\n  \\displaystyle i\\partial_tu+(-\\Delta)^{\\frac{a}{2}}u=0, \\quad 0<a<1,\n\\end{equation*}\nof fractional \nSchr\\\" odinger equations in ${\\mathbb R}^n\\times {\\mathbb R}^+$.\nHowever, observe that, in contrast to the classical Schr\\\"{o}dinger equation, \nthe phase velocity for \\eqref{fracS} becomes arbitrarily large in the long wave limit but is bounded (and actually tends to zero) in the short wave limit.\n\\end{remark}\n\nA natural extension of DBU for linear  Schr\\\"odinger equations is the\ninitial-value problem  in the presence of \nexternal potentials $V(x,t)\\in {\\mathbb R}$.  While we are not \ngoing to deal here with this issue in full generality, we note that an \nimmediate consequence of Remark \\ref{amp} is the appearance of dispersive blow up for Schr\\\"odinger equations with a Stark potential, {\\it i.e.}\n\\begin{equation}\\label{eq:stark}\ni \\partial_t v + \\Delta v - (E\\cdot x)v = 0, \\quad\\,\\, v\\big|_{t=0} = v_0(x),\n\\end{equation}\nwhere $E\\in {\\mathbb R}^n$. This equation models electromagnetic wave propagation in a constant electric field. Solutions of \\eqref{eq:stark} \nare connected to solutions of the free Schr\\\"odinger equation through the Avron-Herbst formula \\cite{AH}. Indeed, it is easy to check that if $v$ solves \n\\eqref{eq:stark}, then \n\\[\nu(x,t) = v\\left(x +  t^2  E, t\\right) e^{-i tE\\cdot x -i  \\frac{t^3}{3}|E|^2},\n\\]\nsolves the linear problem \\eqref{eq:linSch} with the same initial data. Thus, if the  $u$ \nthat solves the free Schr\\\"odinger equation \nexhibits dispersive blow up at a given $(x_*,t_*)$, then so does the solution $v$ of \\eqref{eq:stark} at the point $(x_*+2t_*^2E,t_*)$.\n\n\nAn analogous result is also true in the case of linear Schr\\\"odinger equations with \nisotropic quadratic potentials, {\\it viz.}\n\\begin{equation}\\label{eq:linharm}\ni\\partial_tu + \\Delta u \\pm \\omega^2 |x|^2 u=0, \\quad u\\big|_{t=0} = u_0(x),\n\\end{equation}\nwhere $\\omega \\in {\\mathbb R}$. The two signs correspond, respectively,\n to attractive ($-$) and repulsive ($+$) potentials. \nFollowing \\cite{Ca}, we find that if $v$ solves \\eqref{eq:linharm} with attractive harmonic potential, then \n\\[\nu(x,t) = \\frac{1}{(1+(2\\omega t)^2)^{n\/4}} \\, v \\left(\\frac{\\arctan (2\\omega t) }{\\omega}, \\frac{x}{\\sqrt{1+ \n(2\\omega t) ^2}}\\right) e^{ i  \\frac{ 2\\omega^2  x^2 t}{(1+(2\\omega t)^2}},\n\\]\nsolves \\eqref{eq:linSch}. Thus, dispersive blow up for $u$ again implies dispersive blow up for $v$, although at a \nshifted point in  space-time.\nA similar formula is available in the repulsive case (see \\cite{Ca2}). \n\n\\begin{remark}  Alternatively, one can prove dispersive blow up \nfor  linear Schr\\\"odinger equations with quadratic potentials \nusing (generalized) Mehler formulas for the associated Schr\\\"odinger group.\nThe  Mehler formulas for solutions are \n\\[\nu(x,t)=e^{-in\\frac{\\pi}{4} \\text{sgn}\\; t} \\left|\\frac{\\omega}{2\\pi \\sin \\omega t}\\right|^{\\frac{n}{2}}\\int_{{\\mathbb R}^n} e^{\\frac{i\\omega}{\\sin \\omega t}(\\frac{|x|^2+|y|^2}{2} \\cos \\omega t-x\\cdot y)}\\, u_0(y)\\, dy,\n\\]\n for \\eqref{eq:linharm} in the attractive situation, while in the repulsive case, one has\n\\[\nu(x,t)=e^{-in\\frac{\\pi}{4} \\text{sgn}\\; t} \\left|\\frac{\\omega}{2\\pi \\sinh \\omega t}\\right|^{\\frac{n}{2}}\\int_{{\\mathbb R}^n} e^{\\frac{i\\omega}{\\sinh \\omega t}(\\frac{|x|^2+|y|^2}{2} \\cosh \\omega t-x\\cdot y)}\\, u_0(y) \\, dy\n\\]\n(see  \\cite{Ca2} again).  In the attractive case, Mehler's formula is valid for $|t|<\\frac{\\pi}{2\\omega}$, while in the repulsive case it makes sense for all $t>0$. \nGeneralizations of these formulas are available and allow one to infer that dispersive blow up\ncan occur in the presence  of  anisotropic \nquadratic potentials (see \\cite{Ca2}).\n\\end{remark}\n\nWe close this subsection by noting, that at least for $n=1$, it is easy to show that dispersive blow up is stable under the influence of a \nrather general class of external (real-valued) potentials $V\\in C({\\mathbb R}; L^2({\\mathbb R}))$. To this end, consider the linear\nSchr\\\"odinger equation\n\\begin{equation}\\label{eq:linVSch}\ni \\partial_t u + \\partial^2_{x} u - V(x,t) u=0, \\quad u\\big|_{t=0} = u_0(x),\n\\end{equation}\nwhich which can rewritten, using Duhamel's formula, as\n\\begin{equation}\\label{eq:Vduhamel}\nu(x,t) = e^{i t \\partial_x^2} u_0(x)- i \\int_0^t e^{i (t-s) \\partial _x^2} V(s,x) u(x,s) \\, ds =: e^{i t \\Delta} u_0(x) - i I_{V} (x,t) .\n\\end{equation}\nIn view of \\eqref{eq:group}, we have  formally that\n\\begin{equation}\\label{eq:I}\nI_V(x,t)=\\frac{1}{(4i \\pi t)^{1\/2}} \\int_0^t\\int_{{\\mathbb R}^n}\\frac{1}{(t-s)^{\\frac{1}{2}}}\\exp \\left(i\\frac{|x-y|^2}{4(t-s)}\\right) V(s,y)u(y,s) \\, dy \\, ds.\n\\end{equation}\nNow, assume that the first term on the right-hand side of \\eqref{eq:Vduhamel} exhibits dispersive blow up at some $(x_*, t_*)$.   \nThen, it suffices to show that $I_V(x,t)$ is continuous for $(x,t)\\in {\\mathbb R}^n\\times [0,T]$, for some $T>t_*$ to conclude that the solution of the \\eqref{eq:linVSch} \nexhibits dispersive blow up at the same point $(x_*, t_*)$.  If it is established that $I_V(x,t)$ \nis locally bounded as a function of $x$ and $t$ in the range ${\\mathbb R}^n\\times [0,T]$ for some\n$T > t_*$, then Lebesgue's dominated convergence \ntheorem will imply the desired continuity.  \nTo show local boundedness,  first apply the Cauchy-Schwartz inequality to find\n\\begin{equation}\\label{bound}\n| I_V(x,t) | \\leq \\frac{1}{(4\\pi |t|)^{1\/2}} \\int_0^t \\frac{1}{|t-s|^{\\frac{1}{2}}} \\, \\| V(\\cdot, s) \\|_{L^2} \\| \\| u_0 \\|_{L^2} \\, ds,\n\\end{equation}\nwhere conservation of mass, $\\| u(\\cdot, t)\\|_{L^2} = \\|u_0 \\|_{L^2}$, has been used. \nDue to our assumption on $V$ \nand the fact that $t\\mapsto t^{-1\/2}$ is locally integrable, the right-hand side of \\eqref{bound} is finite and we are done.\n\\vspace{.1cm}\n\nA similar argument will be used preently in the study of DBU for the\n nonlinear Schr\\\"odinger equations in dimension $n=1$. \nIt is clear, however, that in general dimensions $n>1$ a more refined analysis is needed.\n\n\n\n\n\\subsection{Smoothing properties of the free Schr\\\"odinger group}\\label{sec:smooth}\n\nIn this subsection, some  technical results on the smoothing properties of the \nfree Schr\\\"odinger group $S(t) = e^{i t \\Delta}$ are reviewed. They will find use\nin Section 4.   \n\nFirst, recall the notion of admissible index-pairs.\n\\begin{definition} \\label{def:adm} The pair $(p,q)$ is called {\\it admissible} if\n\\begin{equation*}\n\\frac{2}{q}=\\frac{n}{2}-\\frac{n}{p}, \\quad \\text{and} \\ \n    \\left\\lbrace\n    \\begin{array}{l}\n    2\\leq p<\\frac{2n}{n-2},\\ \\text{for} \\, n\\geq 3, \\\\\n    2\\leq p<+\\infty,\\ \\text{if} \\ n=2,\\\\\n    2\\leq p\\leq +\\infty,\\ \\text{if} \\ n=1.\n    \\end{array}\\right.\n\\end{equation*}\n\\end{definition} \n\nFrom now on, for any index $r>0$,  its H\\\"older dual is denoted $r'$, {\\it i.e.} $\\frac{1}{r}+\\frac{1}{r'} = 1$.\nThe well known Strichartz estimates for the Schr\\\"odinger group $S(t) =  e^{i t \\Delta}$ are recounted \nin the next lemma (see \\cite{Caz, LP} for more details).\n\n\\begin{lemma} \\label{lem:Strich}\nIf $(p,q)$ is admissible, then the group $\\{ e^{i t \\Delta} \\}_{t\\in {\\mathbb R}}$ satisfies\n\\[\n\\left(\\int_{-\\infty}^\\infty \\big\\| e^{i t \\Delta} f  \\big\\|^q_{L^p({\\mathbb R}^n)} \\, d t\\right)^{\\frac{1}{q}} \\leq C \\big\\| f \\big\\|_{L^2({\\mathbb R}^n)} \n\\]\nand \n\\[\n \\left(\\int_{-\\infty}^\\infty \\Big \\| \\int_{0}^t e^{i (t-s) \\Delta} g(\\cdot, s)  \\, ds \\Big \\|^q_{L^p({\\mathbb R}^n)} \\, d t\\right)^{\\frac{1}{q}} \\! \\leq \n\\, C  \\left( \\int_{-\\infty}^\\infty \\big\\| g(\\cdot, t)  \\big\\|^{q'}_{L^{p'}({\\mathbb R}^n)} \\, \nd t \\right)^{\\frac{1}{q'}}\\!, \n\\]\nwhere $C = C(p,n)$.\n\\end{lemma}\nThe estimates stated above can be interpreted as global smoothing properties of the free Schr\\\"odinger group $S(t)$. \nIn addition to that, $S(t)$ is known to also induce local smoothing effects, some of which are collected in the following lemma (for proofs, see \\cite[Chapter 4]{LP}).\nFor $1 \\leq j \\leq n$, denote the so-called homogenous derivatives of order $s>0$ by\n\\begin{equation}\\label{eq:homD}\n\\begin{split}\n&D^s_{x_j} f(x):= \\mathcal F^{-1}  \\big(|\\xi_j|^s \\widehat f(\\xi)\\big)(x) \\quad \\text{and, for $n=1$,} \\\\\n&D^s f(x):= \\mathcal F^{-1}  \\big(|\\xi|^s \\widehat f(\\xi)\\big)(x).\n \\end{split}\n\\end{equation}\n\\begin{lemma} \\label{lem:smooth} \nIf $n=1$ and $f \\in L^2({\\mathbb R})$, then\n\\[ \n\\sup_{x\\in {\\mathbb R}} \\ \\int_{\\mathbb R}  \\big| D_{x}^{1\/2}e^{it \\partial^2_x } f (x) \\big|^2 \\, dt \\leq C\\|f\\|^2_{L^2({\\mathbb R})}.\\]\nLet $n\\ge 2$. Then, for all $j\\in \\lbrace 1,\\dots ,n\\rbrace$ and $f \\in L^2({\\mathbb R}^n)$, \n\\[\n\\sup_{x_j\\in {\\mathbb R}} \\ \\int_{{\\mathbb R}^n}  \\big| D_{x_j}^{1\/2}e^{it\\Delta} f (x)\\big|^2 dx_1\\dots dx_{j-1}dx_{j+1}\\dots dx_n  dt  \\leq C\\|f\\|^2_{L^2({\\mathbb R}^n)}.\n\\]\n\\end{lemma}\n\nHelpful inequalities  involving the\nSchr\\\"odinger maximal function\n\\[\nS_T^*f(x):= \\sup_{0\\leq t\\leq T} |e^{it\\Delta}f(x)|\n\\]\nare derived in  \\cite {RV} and  \\cite{V}.  They are reported in the next lemma.\n\\begin{lemma}\\label{lem:B}  The inequality\n\\begin{equation}\\label{max}\n\\| S_T^*f\\|_{L^q({\\mathbb R}^n)} \\leq C_T \\|f\\|_{H^\\sigma({\\mathbb R}^n)}\n\\end{equation}\nholds if either\n\\begin{equation}\\label{5.2}\nn=1\\quad \\text{with}\\quad\\begin{cases}\nq>2\\quad\\text{and}\\quad \\sigma \\geq \\max\\lbrace \\frac{1}{q}, \\frac{1}{2}-\\frac{1}{q}\\rbrace, \n \\\\\nq=2\\quad\\text{and}\\quad \\sigma >\\frac{1}{2},\n\\end{cases}\n\\end{equation}\nor  \n\\begin{equation}\\label{5.2bis}\nn>1\\quad \\text{with}\\quad\\begin{cases}\nq\\in (2+\\frac{4}{(n+1)},\\infty)\\quad\\text{and}\\quad \\sigma >n(\\frac{1}{2}-\\frac{1}{q}),\\\\\nq\\in[2,2+\\frac{4}{(n+1)}]\\quad\\text{and}\\quad \\sigma >\\frac{3}{q}-\\frac{1}{2}.\n\\end{cases}\n\\end{equation}\n\\end{lemma}\n\n\nWith these results at hand, attention is turned to establishing\n dispersive blow up for nonlinear Schr\\\"odinger equations.\n\n\n\n\n\n\\section{Dispersive blow up for nonlinear Schr\\\"odinger equations }\\label{sec:NLS}\n\nIn this section, the initial-value problem\n\\begin{equation}\\label{eq:NLS}\ni\\partial_t u+\\Delta u\\pm |u|^p u=0,\\quad u\\big|_{t=0}=u_0(x),\n\\end {equation}\nfor the nonlinear Schr\\\"odinger equation is considered.  Here,  $x\\in {\\mathbb R}^n$ and $p>0$ is not necessarily an integer. \nFinite-time dispersive blow up was established for $n=1$ and $p\\in (0,3)$ in \\cite{BS2}. \nOur strategy to improve upon this result relies upon the theory developed in  \\cite{CW}, where the Cauchy problem \\eqref{eq:NLS} \nwas studied for $u_0\\in H^s({\\mathbb R}^n)$ for various values of $s$.  \n\n\n\\subsection{Local well-posedness in $H^s$}\\label{sec:LWP}\n\nFor $1\\le r < \\infty$ and $s>0$, define\n\\[\nH^{s,r}({\\mathbb R}^n)=\\Big\\{ f\\in L^r({\\mathbb R}^n): \\mathcal F^{-1}\\big[ (1+|\\xi|^2)^{s\/2} \\widehat f(\\xi) \n \\big] \n\\in L^r({\\mathbb R}^n) \\Big\\}.\n\\] \nThese are the  standard Bessel potential spaces. According to \\cite{St}, which uses the notation $L^{s,r}$ instead of $H^{s,r}$, these spaces may be characterized in the following manner.\nLet $s\\in (0,1)$ and $\\frac{2n}{(n+2s)}<r<\\infty$. Then\n$f\\in H^{s,r}({\\mathbb R}^n)$ if and only if $f\\in L^r({\\mathbb R}^n)$ and \n\\begin{equation}\\label{eq:frac}\n\\mathcal D^s f(x)=\\left(\\int_{{\\mathbb R}^d} \\frac{|f(x)-f(y)|^2}{|x-y|^{n+2s}} \\, dy\\right)^{1\/2}\\in L^r({\\mathbb R}^n).\n\\end{equation}\nThe space $H^{s,r}({\\mathbb R}^n) \\equiv (I-\\Delta)^{-s\/2}L^r({\\mathbb R}^n)$ is equipped with the norm \n \\begin{align*}\n \\|f\\|_{H^{s,r}({\\mathbb R}^n)}=\\|(I-\\Delta)^{s\/2}f\\|_{L^r({\\mathbb R}^n)} =  & \\ \\|f\\|_{L^r({\\mathbb R}^n)}+\\|\\mathcal D^s f\\|_{L^r({\\mathbb R}^n)} \\\\\n \\simeq  & \\ \\|f\\|_{L^r({\\mathbb R}^n)}+\\| D^s f\\|_{L^r({\\mathbb R}^n)},\n\\end{align*} \nwhere $D^s$ is the homogenous derivative defined in \\eqref{eq:homD}. \nObserve that, for $r=2$, $H^{s,2}({\\mathbb R}^n)\\equiv H^{s}({\\mathbb R}^n)$, the usual $L^2$--based Hilbert space. \nIn addition, a straightforward calculation reveals that \n\\begin{equation}\\label{3}\n\\|\\mathcal D^s f\\|_{L^2({\\mathbb R}^n)}=c_n\\||\\xi |^s \\widehat f \\, \\|_{L^2({\\mathbb R}^n)} \\equiv c_n \\| D^s f\\|_{L^2({\\mathbb R}^n)},\n\\end{equation} \nwith $c_n$  a constant depending only on the dimension $n$.  \nIf $(q,r)$ is an admissible pair as defined in Section 2, then the space \n\\begin{equation} \\label{solutionspace}\nW^T_{s,n}\\, = \\, C\\big([0,T];H^s({\\mathbb R}^n)\\big)\\cap L^q\\big([0,T|;H^{s,r}({\\mathbb R}^n)\\big) \n\\end{equation}\nwill also appear.   Of course, these spaces depend  on the admissible pair $(q,r)$, \nbut this dependence is suppressed in the notation.  \nThe following local well-posedness theorem established in \\cite{CW} makes use of both the\nBessel-potential spaces and the latter, spatial-temporal spaces.\n\\begin{proposition}\\label{prop:CW}\nLet $s>s_{p,n}=\\frac{n}{2}-\\frac{2}{p},$ $s>0,$ with $s$ otherwise arbitrary if $p$ is \nan even integer, $s < p+1$ if $p$ is an odd integer and  \n$\\lfloor s \\rfloor <p$ if $p$ is not an integer. For given initial data $u_0\\in H^s({\\mathbb R}^n)$, \n\\begin{enumerate}\n\\item there exist $T=T(\\|u_0\\|_{H^s})>0$ and a unique solution\n\\[\nu\\in C\\big([0,T];H^s({\\mathbb R}^n)\\big)\\cap L^q\\big([0,T|;H^{s,r}({\\mathbb R}^n)\\big)\\equiv W^T_{s,n}\n\\]\nfor all  pairs $(q, r)$ admissible in the sense of Definition \\ref{def:adm}, \n and\n\\item the local existence time $T=T(\\|u_0\\|_{H^s}) \\to +\\infty$, as $\\|u_0\\|_{H^s({\\mathbb R}^n)}\\to 0.$\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{remark}The proof of this result follows  from a fixed point argument based on the Strichartz estimates displayed in Lemma \\ref{lem:Strich}.  Recall that \nthe notation $\\lfloor s \\rfloor$ connotes the largest integer less than or equal to $s$. \\end{remark}\n\n\nTo apply Proposition \\ref{prop:CW} in our context, we need to show that the class of initial data constructed in Section \\ref{sec:lin} \n(yielding dispersive blow up for the free Schr\\\"odinger evolution) admits  Sobolev \nclass regularity that will allow the use of Lemma \\ref{prop:CW}. \n Via scaling and translation, dispersive blow up can be achieved at any point $(x_*, t_*)$ in \nspace-time, so without loss of generality, fix $(x_*, t_*) = (0, 1)$ and focus upon the \ninitial data\n\\begin{equation}\\label{eq:IC}\nu_0(x) = \\frac{e^{- 4 i |x |^2}}{(1+|x|^2)^m}\\ ,\\quad \\text{with $\\frac{n}{4} < m \\le \\frac{n}{2}$.}\n\\end{equation}\nThis initial value $u_0$ has Sobolev regularity explained in the next lemma.\n\\begin{lemma}\\label{lem:linear}\nLet  $u_0$ be as depicted  in \\eqref{eq:IC}. Then\n$u_0\\in H^s({\\mathbb R}^n)$ if \\;$2m>s+\\frac{n}{2}$. In particular, if $m=\\frac{n}{2},$   \nthen $u_0\\in H^s({\\mathbb R}^n)$ for any $ s\\in (0,\\frac{n}{2})$, whereas if $m=\\frac{n}{4}^+,$ then  $s \\in (0,0^+)$.\n\\end{lemma}\n\n\\begin{proof} Consider first the case $0<s<1$.  \nFor $s$ in this range, Propositions 1 and 2 in \\cite{NP} provide the inequalities \n\\[\n| \\mathcal D^s e^{i|x|^2}|\\leq c_n(1+|x|^s)\n\\]\nand\n\\[\n\\|\\mathcal D^s(fg)\\|_{L^2({\\mathbb R}^n)}\\leq \\|f\\mathcal D^s g\\|_{L^2({\\mathbb R}^n)}+\\|g\\mathcal D^s f\\|_{L^2({\\mathbb R}^n)},\n\\]\nwhere $\\mathcal D^s$ is defined in \\eqref{eq:frac}.\nCombining these estimates with identity \\eqref{3} and using interpolation, one arrives \nat the inequality\n\\begin{align*}\n&\\, \\left\\Vert \\mathcal D^s \\left(\\frac{e^{i|x|^2}}{(1+|x|^2)^m}\\right)\\right\\Vert_{L^2({\\mathbb R}^n)} \\\\ \n &\\, \\leq  \\left\\Vert\\frac{1}{(1+|x|^2)^m}\\mathcal D^s(e^{i|x|^2})\\right\\Vert _{L^2({\\mathbb R}^n)}+\\left\\Vert \\mathcal D^s\\left(\\frac{1}{(1+|x|^2)^m}\\right)\\right\\Vert_{L^2({\\mathbb R}^n)}\\\\\n&\\, \\leq c_n \\left\\Vert \\frac{1}{(1+|x|^2)^m}\\right\\Vert_{L^2({\\mathbb R}^n)}+c_n \\left\\Vert \\frac{|x|^s}{(1+|x|^2)^m}\\right\\Vert_{L^2({\\mathbb R}^n)}\\\\\n&\\quad \\  +\\left\\Vert \\frac{1}{(1+|x|^2)^m}\\right\\Vert_{L^2({\\mathbb R}^n)}^{1-s} \\left\\Vert D\\left(\\frac{1}{(1+|x|^2)^m}\\right)\\right\\Vert^s_{L^2({\\mathbb R}^n)}.\n \\end{align*}\nThe right-hand side of this inequality is finite if and only if $2m-s>\\frac{n}{2}.$  \n\nIf $s$ is a positive integer, the result follows from Leibnitz' rule. \nIf instead, $s = s' +k$ where $0 < s' < 1$, then simply apply the above analysis \nto the $k^{\\rm th}$--deriviative $u_0^{(k)}$.    \n\\end{proof}\n\nNotice that if $m = \\frac{n}{2}$, we can certainly choose the value $s$ in the interval $(\\frac{n}{2}-\\frac{2}{p},\\frac{n}{2}]$ where Proposition  \\ref{prop:CW}  applies.   \n\n\n\n\n\\subsection{Proof of dispersive blow up for nonlinear Schr\\\"odinger equations}\\label{sec:DBUNLS}\n\nHere is the detailed statement of dispersive blow up for the initial-value problem \\eqref{eq:NLS}.\nIn this theorem, the \nLebesgue index $p\\geq \\lfloor \\frac{n}{2} \\rfloor$ if $p$ is not an even integer.\n\n\\begin{theorem}\\label{thm:main}\nGiven $t_{*}>0$ and $x_*\\in {\\mathbb R}^n,$ for any $s\\in (\\frac{n}{2}-\\frac{1}{2p},\\frac{n}{2}]$,\n there are initial data  $u_0\\in H^s({\\mathbb R}^n)\\cap L^{\\infty}({\\mathbb R}^n)\\cap C^{\\infty}({\\mathbb R}^n)$ \nand a $T = T(\\|u_0\\|_{H^s})>t_*$\n such that \n\\begin{enumerate}   \n\\item the  initial-value problem   \\eqref{eq:NLS} has a unique solution $u$ \nin the class described in Proposition   \\ref{prop:CW} which is defined \nat least on  the time interval $[0,T]$,   and\n\\item $u$ exhibits dispersive blow up, which is to say,\n$$ \\lim_{{(x,t)\\in {\\mathbb R}^n\\times [0,T] \\to (x_*,t_*)}} |u(x,t)|=+\\infty.$$\n\\item Moreover, $u$ is a continuous function of $(x,t)$ on ${\\mathbb R}^n\\times( [0,T])\\setminus \\lbrace t_*\\rbrace)$ and\n\\item $u(\\cdot,t_*)$ is a continuous function of $x$ on ${\\mathbb R}^n\\setminus \\lbrace x_* \\rbrace.$\n\\end{enumerate}\n\\end{theorem}\n\nThis theorem extends the results of \\cite{BS2} to the cases where $n\\geq 2$ and $p\\geq 3.$ \nNotice that the nonlinearity $y \\mapsto |y|^py$ is smooth when $p$ is an even integer.  \nOtherwise, it has finite regularity and hence the restriction on $p$ in those cases.\n\n\n\\begin{proof} The proof is provided in detail  for $p>0$ in the case $n=1$ and, when $n\\geq 2$, \nfor the case $p=2k$,\n$k$ a positive integer.  It \nwill be clear from the argument that the result extends to the case of \n $p\\geq \\lfloor \\frac{n}{2} \\rfloor$ if $p$ is not an even integer. \n\nAs already mentioned, we may assume that the  dispersive blow up for the \nfree Schr\\\"odinger group $S(t)$ \noccurs at $x_* = 0$ and $t_*=1$. Note that the same is true for initial data of the form $\\delta u_0$, where $u_0$ is \nas in \\eqref{eq:IC} with $m = \\frac{n}{2}$, say, $\\delta > 0$  arbitrary and $s$ satisfying \nthe conditions in Proposition \\ref{prop:CW}. \nIn view of  part (3) of the latter proposition,  the local existence time $T^* = T(\\| \\delta u_0\\|_{H^s})>0$ can be made arbitrarily large by choosing $\\delta$  sufficiently small \nand hence we can always achieve $T^*>1 = t_* $.\\\\\n\n{\\it Step 1.}  Take as initial \ndata $\\delta u_0$, where $u_0$ is as in \\eqref{eq:IC} with $m = \\frac{n}{2}$.  Let $s \\in (\\frac{n}{2} - \\frac{2}{p}, \\frac{n}{2}]$ with $p \\geq \\lfloor\\frac{n}{2}\\rfloor$.   Then \n$s$ satisfies the conditions of Proposition \\ref{prop:CW}.  As noted above, by choosing \n$\\delta$ small enough, we can be sure that the solution $u$ of \\eqref{eq:NLS} \nemanating from $u_0$ \nexists and is unique in $C([0,T]:H^s)$ where $T > t_* = 1$.   \n\nDuhamel's formula allows us to represent $u$ in the form\n\\begin{equation}\\label{eq:duhamel}\nu(x,t) = e^{i t \\Delta} u_0(x) \\pm i \\int_0^t e^{i (t-s) \\Delta} |u(x,s)|^p u(x,s) \\, ds =: e^{i t \\Delta} u_0(x) \\pm i I (x,t) ,\n\\end{equation}\nwhere, at least formally, $I(x,t)$ can be written as the double integral\n\\begin{equation}\\label{eq:I}\nI(x,t)=\\frac{1}{(4i \\pi t)^{n\/2}} \\int_0^t\\int_{{\\mathbb R}^n}\\frac{1}{(t-s)^{\\frac{n}{2}}}\\exp \\left(i\\frac{|x-y|^2}{4(t-s)}\\right) |u(y,s)|^p  u(y,s) \\, dy \\, ds.\n\\end{equation}\nThe first term on the right-hand side of \\eqref{eq:duhamel} \n exhibits dispersive blow up at $(x_*, t_*) = (0,1)$ on account of the choice of $u_0$.   \n If it turns out that  $I$ is continuous for $(x,t)\\in {\\mathbb R}^n\\times [0,T]$, then it is \nimmediately concluded that \\eqref{eq:duhamel} (and thus \\eqref{eq:NLS}) exhibits \ndispersive blow up at the same point $(x_*, t_*)=(0,1)$.  To show that $I(x,t)$ \nis continuous, it suffices to prove that it is locally bounded as a function of $x$ and $t$, \nsince then  Lebesgue's dominated convergence theorem will imply $I$ is continuous on ${\\mathbb R}^n\\times [0,T]$. \\\\\n\n{\\it Step 2.}  Consider  $n=1$ first, since  a more direct proof can be made \nin this case.  \nThe  initial data is \n\\[\nu_0(x)= \\, \\frac{\\delta e^{-i4 x^2}}{(1+x^2)^{\\frac{1}{2}}},\\quad x\\in {\\mathbb R}.\n\\] \nThe function $u_0$ lies in $H^s({\\mathbb R})$ for any $s$ in the range  $0\\leq s<\\frac{1}{2}.$ \nProposition \\ref{prop:CW} then provides a local in time solution $u \\in C([0,T]; H^s({\\mathbb R}))$ to the Cauchy problem \\eqref{eq:NLS} provided that \n\\[\n0<s<\\frac{1}{2} \\quad \\,\\, \\text{and} \\,\\, \\quad 0<p\\leq \\frac{4}{1-2s}.\n\\]\nAs mentioned already, $T$ may be taken larger than 1 by choosing $\\delta$ small.  By Sobolev imbedding, $H^s({\\mathbb R})\\subset L^{r+1}({\\mathbb R})$ if $\\frac{1}{r+1}\\geq \\frac{1}{2}-s.$\nHence, for  $s=\\frac{1}{2}-\\varepsilon$, where $\\varepsilon>0$ is fixed and small, it is inferred  \nthat $u\\in C([0,T]; L^{r+1}({\\mathbb R}))$ for all $r$ in the range \n\\[\n0<r\\le \\min \\left\\{\\frac{2}{\\varepsilon}, \\frac{1-\\varepsilon}{\\varepsilon}\\right\\}=\\frac{1-\\varepsilon}{\\varepsilon}. \n\\]\nAs $\\epsilon > 0$ was arbitrary, it  follows that  \n$u(\\cdot, t)\\in L^{r+1}({\\mathbb R})$, for  $r\\geq 1$ arbitrarily large. \nIn consequence, $I(x,t)$ is locally bounded.  \nIndeed, using H\\\"older's inequality, it is seen that\n\\begin{align*}\n|I(x,t)| \\le &\\ \\int_0^t \\frac{1}{(t-s)^{1\/2}} \\, \\big\\| u(\\cdot, s)\\big\\|_{L^{p+1}}^{p+1} \\, ds \\\\\n\\le &\\  \\left(\\int_0^t \\frac{1}{(t-s)^{\\gamma\/2}} \\, ds \\right)^{\\frac{1}{\\gamma} }\n \\left(\\int_0^t \\big\\| u(\\cdot, s)\\big\\|_{L^{p+1}}^{\\gamma' (p+1)} \\, ds \\right)^{\\frac{1}{\\gamma'} }\n\\end{align*}\nwhere $\\frac{1}{\\gamma}+\\frac{1}{\\gamma'}=1$ and $\\gamma \\in (1,2)$. Having in mind that $u\\in C([0,T]; L^{p+1}({\\mathbb R}))$, it transpires that\n\\[ |I(x,t)| \\le C T^{\\frac{1}{\\gamma'}} \\sup_{t\\in[0,T]} \\| u(\\cdot, t) \\|_{L^{p+1}}^{\\gamma'(p+1)},\n\\]\nfor all $x \\in {\\mathbb R}$, which concludes the proof in the case $n=1$.\\\\\n\n\n\n{\\it Step 3.} In  case $n\\ge 2$,  the strategy employed \nabove no longer works beause the factor $t\\mapsto t^{-n\/2}$ appearing \nin the representation formula \\eqref{eq:I} is no longer locally integrable. However, it will be shown \nin Proposition \\ref{prop:key} below that the double integral $I$ is in fact half a derivative \nsmoother than one would naively expect. To make use of this result,  choose initial data $u_0$ \nof the form \\eqref{eq:IC} with $m \\in (\\frac{n}{2}-\\frac{1}{4p},\\frac{n}{2}]$, where \n$p\\ge \\lfloor\\frac{n}{2}\\rfloor$.\nIt is immediately inferred from Lemma \\ref{lem:linear} that \n\\[u_0\\in H^s({\\mathbb R}^n)\\quad \\text{for any} \\,\\, s\\in \\Big( \\frac{n}{2}-\\frac{1}{2p},\\frac{n}{2}\\Big].\\]\nNotice that \n\\[\\frac{n}{2}-\\frac{1}{2p}>\\frac{n}{2}-\\frac{2}{p}=s_{p,n}\\] \nso that Proposition \\ref{prop:CW} applies and therefore $u\\in W^T_{s,n}$ (see \\eqref{solutionspace})\nsatisfies the integral form \\eqref{eq:duhamel} of the nonlinear Schr\\\"odinger equation.  \nProposition \\ref{prop:key} below then shows  that \n\\[\nI \\in C([0,T]; H^{s+1\/2}({\\mathbb R}^n)),\n\\]\nand hence, for $s\\in (\\frac{n}{2}-\\frac{1}{2p},\\frac{n}{2}],$ one has\n\\[\nI \\in C({\\mathbb R}^n\\times [0,T])\\cap L^\\infty({\\mathbb R}^n\\times [0,T]).\n\\]\nThe proof is complete.\n\\end{proof}\n\n\n\n\n\\begin{remark} It was shown in \\cite[Theorem 2.1]{BS2} that dispersive blow up results \nare stable under smooth and localized perturbations of the data.  That is to say,  \nif they hold for data $u_0,$ then they also hold for data $u_0+w$ where, for instance, $w \\in H^\\infty({\\mathbb R}^n).$   In particular the data \nleading to DBU do {\\it not} need to be radially symmetric. The same is true for Theorem \\ref{thm:main}.\nThe proofs of these results consists of writing the equation satisfied by $w$ and showing\nthat it has bounded, continuous solutions.  The details follow exactly the argument \ngiven already in \\cite{BS2}.  \n\nIn addition, there is  a kind of density of initial data leading to dispersive blow up.  More \nprecisely,  given $u_0\\in H^s({\\mathbb R}^n)$ with $s>n\/2$ and $\\epsilon >0,$ there exists\n$\\phi\\in H^r({\\mathbb R}^n), $ $r\\in \\big(\\frac{n}{2}-\\frac{1}{2p},\\frac{n}{2}\\big]$ with \n\\begin{equation}\n  \\label{density}\n    \\|u_0-\\phi \\|_{H^r({\\mathbb R}^n)}<\\epsilon,\n\\end{equation}\nsuch that the initial data $\\phi$ leads to dispersive blow up in the sense of Theorem \\ref{thm:main}. Indeed, it suffices to take \n$\\phi=u_0+\\delta v_0,$ where $v_0$ leads to dispersive blow up and  $\\delta >0$ is small enough \nthat \\eqref{density} holds. A combination of Theorem \\ref{thm:main} and \n Proposition \\ref{prop:CW} then \nimplies the above assertion.\n\\end{remark}\n\n\n\n\n\n\\section{Global smoothing of the Duhamel term and applications}\\label{sec:global}\n\nIn this section, the proof of Theorem \\ref{thm:main} is completed by showing the Duhamel \nterm in the integral representation \\eqref{eq:duhamel} of the solution  of the initial-value problem \\eqref{eq:NLS}\nis smoother than is the linear term involving only the initial data.  In fact, several\ndifferent results of smoothing by the Duhamel term will be developed, though \nthe first one is enough for the dispersive blow up result in Section 3.  \n\n\\subsection{Smoothing by half a derivative}\n\nThe following proposition\nsuffices to complete the proof \nof Theorem \\ref{thm:main}.  \n\n\\begin{proposition}\\label{prop:key}\nLet $u_0\\in H^s({\\mathbb R}^n),$\\;$s>\\frac{n}{2}-\\frac{1}{2p}$ with $p\\ge 1$ \nand $\\lfloor p+1\\rfloor \\geq s + \\frac12$ if $p$ is not an even integer.\nLet  $u\\in W^T_{s,n}$ be the solution of \\eqref{eq:NLS} satisfying\n\\[\nu(x,t)=e^{it\\Delta}u_0(x)\\pm i \\int_0^te^{i(t-s)\\Delta} |u(x,s)|^pu(x,s)ds=:e^{it\\Delta}u_0(x) \\pm i I(x,t).\n\\]\nThen $I \\in C([0,T]; H^{s+\\frac12}({\\mathbb R}^n)).$ \n\\end{proposition}\nIn other words, the integral term $I$ is \\lq\\lq smoother\" than the free propagator $e^{it\\Delta}u_0$ by \nhalf a derivative.\nThis is  the key point needed in the proof of Theorem \\ref{thm:main} for $n\\ge 2$.   In the \nspecial case wherein\nthe nonlinearity $|u|^pu$ is smooth, so when $p = 2k, k$ an integer,  \nProposition \\ref{prop:keybis} will show that  the Duhamel term is\nalmost one derivative smoother than one would expect.\n\n\\begin{remark}  The fact that\nthe nonlinear integral term in  Duhamel's formula is smoother than the linear one\nin certain circumstances has been used in other  works \non nonlinear dispersive equations. \nFor example, in \\cite{LS} it was employed to give a different \nproof of some of the results obtained in \\cite{BS1}. \nIn \\cite{Bou}, this smoothing effect was applied to deduce  global \nwell-posedness below the regularity index provided by the conservation laws of \nmass and energy.  So far as we are aware, however, the result stated in Proposition \n\\ref{prop:key} has not\npreviously been explicitly written down.\n \\end{remark}\n\n\n\n\n\n\n\n\\begin{proof} The details are provided for the case $p=2k, k\\in {\\mathbb N}.$ \nIt will be clear that the arguments extend to the case \nwhere $p\\geq 1$ is not an even integer, but $p\\ge \\lfloor\\frac{n}{2}\\rfloor$ and $n\\ge2$. \\\\\n\n{\\it Step 1.} In the first step, useful estimates on $e^{it\\Delta}u_0$ are derived. \nStart by fixing a $j\\in \\lbrace 1,\\dots,n\\rbrace$ and noticing that\n\\begin{equation}\\label{compA}\n\\sup_{0\\leq t\\leq T}\\left\\{ \\sup_{x_1\\dots x_{j-1}x_{j+1}\\dots x_n}\\Big\\{|e^{it\\Delta} u_0(x)|\\Big\\} \\right\\} \\lesssim \n\\sup_{x_1\\dots x_{j-1}x_{j+1}\\dots x_n}\\left\\{|e^{it(x_j)\\Delta}u_0(x)|\\right\\}\n\\end{equation}\nfor some $t(x_j)\\in [0,T]$.  For  $q\\ge 2$ and $s>\\frac{(n-1)}{q}$, it follows by Sobolev embedding that\n\\begin{equation}\\label{comp}\n\\begin{aligned}\n\\sup_{x_1\\dots x_{j-1}x_{j+1}\\dots x_n}|e^{it(x_j)\\Delta}u_0(x)| & \\, \\lesssim  \n\\big\\|e^{it(x_j)\\Delta}u_0\\big\\|_{H^{s,q}_{x_1\\dots x_{j-1}x_{j+1}\\dots x_n}\\left({\\mathbb R}^{n-1}\\right)} \\\\\n&\\, =c\\big\\|e^{it(x_j)\\Delta}J^s_{\\hat \\jmath}u_0\\big\\|_{L^q_{x_1\\dots x_{j-1}x_{j+1}\\dots x_n}\\big({\\mathbb R}^{n-1}\\big)}\\\\\n&\\, \\lesssim \\big\\|\\sup_t |e^{it\\Delta}J^s_{\\hat \\jmath}u_0|\\;\n\\big\\|_{L^q_{x_1\\dots x_{j-1}x_{j+1}\\dots x_n} \\left({\\mathbb R}^{n-1}\\right)}\n\\end{aligned}\n\\end{equation}\nwhere here and in the following, \n\\[\nJ^s_{\\hat \\jmath }=(1-(\\partial^2_{x_1}+\\dots+\\partial^2_{x_{j-1}}\n+\\partial^2_{x_{j+1}}+\\dots+\\partial^2_{x_n}))^{s\/2}\n\\]\nis defined via the associated Fourier symbol and the subscripts on the function spaces \nindicate which variables participate in the norm. \nUsing \\eqref{compA} and \\eqref{comp} together with \nthe estimates on the Schr\\\"odinger maximal function given in Lemma \\ref{lem:B}, \nthere appears the inequality\n\\begin{equation}\\label{ouf}\n\\begin{aligned}\n&\\big\\|e^{it\\Delta}u_0\\big\\|_{L^q_{x_j}({\\mathbb R};L^\\infty_{x_1\\dots x_{j-1}x_{j+1}\\dots x_n t}({\\mathbb R}^{n}\\times[0,T]))}\\\\\n&\\hspace{1cm}= \\Big\\|\\sup_{0\\leq t\\leq T}\\sup_{x_1\\dots x_{j-1}x_{j+1}\\dots x_n}|e^{it\\Delta}u_0|\\;\\Big\\|_{L^q_{x_j}({\\mathbb R})}\\\\\n&\\hspace{1cm} \\leq C_T\\Big\\|\\sup_{0\\leq t\\leq T}|e^{it\\Delta}J^s_{\\hat \\jmath}u_0|\\;\\Big\\|_{L^q({\\mathbb R}^n)}\\\\\n&\\hspace{1cm} \\lesssim \\big\\|u_0\\big\\|_{H^{\\sigma+s}({\\mathbb R}^n)},\n\\end{aligned}\n\\end{equation}\nwith $q\\geq 2$, $s>\\frac{n-1}{q}$ and $\\sigma>0$ as specified in Lemma \\ref{lem:B}. \nThe inequality \\eqref{ouf}, together with the local smoothing estimates stated in Lemma \\ref{lem:smooth}, \nreveal that\n\\begin{equation}\\label{ouf2}\n\\big\\|D^{\\theta\/2}_{x_j}e^{it\\Delta} u_0\\big\\|_{L_{x_j}^{q\/(1-\\theta)}({\\mathbb R};L^{2\/\\theta}_{x_1\\cdot\\cdot x_{j-1}x_{j+1}\\cdot\\cdot x_nt}({\\mathbb R}^{n-1}\\times[0,T]))} \n\\leq c_T\\big\\|u_0\\big\\|_{H^{(1-\\theta)(\\sigma+s)}({\\mathbb R}^n)},\n\\end{equation}\nwhere $\\theta\\in[0,1]$, $q\\geq 2$, $s>\\frac{n-1}{q}$ and $\\sigma>0$ as before. \\\\\n\n\n{\\it Step 2.}  To bound $\\|D_{x_j}^{s+1\/2} I\\|_{L^\\infty_t([0,T]:L^2({\\mathbb R}^n))}$ above,  let  $j\\in \\lbrace 1,\\dots , n\\rbrace$, $p=2k$ and write\n\\begin{equation}\n\\label{tterm}\n\\begin{aligned}\n&\\Big \\|D_{x_j}^{s+1\/2}  \\int_0^t e^{i(t-s)\\Delta}(|u|^{2k}u)(s) \\, ds \\Big \\|_{L^\\infty([0,T];L^2({\\mathbb R}^n))}\\\\\n&\\hspace{1cm} \\leq \\, c\\big\\|D_{x_j}^{s+1\/2} (|u|^{2k} u)\\big\\|_{L^1([0,T];L^2({\\mathbb R}^n))}\\\\\n&\\hspace{1cm} \\leq \\, cT^{1\/2}\\big\\|D_{x_j}^{s+1\/2} (|u|^{2k} u)\\big\\|_{L^2([0,T]\\times {\\mathbb R}^n)}.\n\\end{aligned}\n\\end{equation}\n To estimate the right-hand side of \\eqref{tterm}, the calculus of inequalities involving fractional derivatives \nderived  in \\cite{KPV93} is helpful.  More precisely,  the following inequality, \nwhich is a particular case of those proved in \n \\cite[Theorem A.8]{KPV93}, will be used.   Let $\\alpha\\in (0,1), \\,\\alpha_1,\\alpha_2\\in[0,\\alpha]$ with $\\alpha=\\alpha_1+\\alpha_2$ and  let $p_1,p_2,q_1,q_2\\in [2,\\infty)$ be such that\n \\[\\frac{1}{2}=\\frac{1}{p_1}+\\frac{1}{p_2}=\\frac{1}{q_1}+\\frac{1}{q_2}.\\] Then\n \\begin{equation}\n \\label{inequality}\n \\begin{aligned}\n &\\big\\| D_{x_j}^{\\alpha}(fg)-fD_{x_j}^{\\alpha}f-g D_{x_j}^{\\alpha}f  \\big\\|_{L^2_{x_j}({\\mathbb R};L^2(Q))}\\\\\n& \\hspace{.7cm}\\leq c\\big\\| D_{x_j}^{\\alpha_1}f\\big\\|_{L^{p_1}_{x_j}({\\mathbb R};L^{q_1}(Q))}   \\big\\|D_{x_j}^{\\alpha_2}g\\big\\|_{L^{p_2}_{x_j}({\\mathbb R};L^{q_2}(Q))},\n \\end{aligned}\n \\end{equation}\n where $Q={\\mathbb R}^{n-1}\\times [0,T]$. \n\nTo illustrate the use of the inequality \\eqref{inequality} in estimating the right-hand side of \\eqref{tterm}, \nassume without loss of generality that $s+\\frac{1}{2}=1+\\alpha$ with  $\\alpha\\in (0,1)$. \nThus, omitting the domains of integration ${\\mathbb R}$ and $Q$,\n\\begin{equation}\n\\label{001}\n\\begin{aligned}\n\\big\\| D_{x_j}^{s+1\/2}(fg)\\big\\|_{L^2_{x_j}L^2}& = \\big\\| D_{x_j}^{1+\\alpha}(fg)\\big\\|_{L^2_{x_j}L^2} \\simeq \\big\\| D_{x_j}^{\\alpha}\\partial_{x_j}(fg) \\big\\|_{L^2_{x_j}L^2}\\\\\n&\\leq c\\Big( \\big\\| D_{x_j}^{\\alpha}(f\\partial_{x_j}g)\\big\\|_{L^2_{x_j}L^2}\n\\big\\| D_{x_j}^{\\alpha}(\\partial_{x_j}f g)\\big\\|_{L^2_{x_j}L^2}\\Big).\n\\end{aligned}\n\\end{equation}\nBy symmetry it suffices to consider only one of the terms \non the right-hand side of \\eqref{001}. From  \\eqref{inequality}, with $\\alpha_1=0$, \nthere obtains\n\\begin{equation*}\n\\label{002}\n\\begin{aligned}\n\\big\\| D_{x_j}^{\\alpha}(g\\partial_{x_j}f)\\big\\|_{L^2_{x_j}L^2}& \\leq  \\big\\| D_{x_j}^{\\alpha}(g\\partial_{x_j}f)-g D_{x_j}^{\\alpha}\\partial_{x_j}f - \\partial_{x_j}fD_{x_j}^{\\alpha}g \\big\\|_{L^2_{x_j}L^2}\\\\\n& \\quad +\\big\\| g D_{x_j}^{\\alpha}\\partial_{x_j}f\\big\\| _{L^2_{x_j}L^2} + \\big\\|  \\partial_{x_j}f D_{x_j}^{\\alpha}g\\big\\| _{L^2_{x_j}L^2}\\\\\n&\\leq \\big\\| g D_{x_j}^{\\alpha}\\partial_{x_j}f \\big\\| _{L^2_{x_j}L^2} + c\\big\\| \\partial_{x_j}f\\big\\|_{L^{p_1}_{x_j}L^{q_1}}   \\big\\|D_{x_j}^{\\alpha_2}g\\big\\|_{L^{p_2}_{x_j}L^{q_2}}\n\\end{aligned}\n\\end{equation*}\nwith $p_1,p_2,q_1, q_2$ restricted as above.\n\n\nUsing the latter inequality to continue the inequality \\eqref{tterm} yields\n\\begin{equation}\\label{KPV}\n\\begin{aligned}\n&\\big\\|D_{x_j}^{s+1\/2} I\\big\\|_{L^\\infty_t([0,T]:L^2({\\mathbb R}^n))}  \\\\ \n &\\hspace{.8cm} \\leq  \\ cT^{1\/2} \\big\\|D_{x_j}^{s+1\/2} (|u|^{2k} u)\\big\\|_{L^2([0,T]\\times {\\mathbb R}^n)}\\\\\n&\\hspace{.8cm} \\leq \\ cT^{1\/2}\\Big ( \\big\\|u\\big\\|^{2k}_{L^{4k}_{x_j}({\\mathbb R}; L^\\infty_{x_1\\dots x_{j-1}x_{j+1}\\dots x_n,t}({\\mathbb R}^{n-1}\\times[0,T]))} \\\\\n& \\hspace{1.5cm} \\times \\big\\|D_{x_j}^{s+1\/2} u\\big\\|_{L^{\\infty}_{x_j}({\\mathbb R}; L^2_{x_1\\dots x_{j-1}x_{j+1}\\dots x_n,t}({\\mathbb R}^{n-1}\\times[0,T]))}+R\\Big),\n\\end{aligned}\n\\end{equation}\nwhere the remainder $R$ includes only estimates for terms involving powers of $u$,  $\\partial_{x_j}u$ and $D_{x_j}^{\\alpha}u$ . These are straightforwardly bounded above \n by use of \\eqref{ouf2}. In fact, a bound for them is an interpolation between the first two terms \non the right-hand side of \\eqref{KPV}. It therefore remains to bound only the terms\n\\begin{equation}\\label{eq:A}\n \\big\\|u\\big\\|^{2k}_{L^{4k}_{x_j}({\\mathbb R}; L^\\infty_{x_1\\dots x_{j-1}x_{j+1}\\dots x_n,t}({\\mathbb R}^{n-1}\\times[0,T]))}\n\\end{equation}\nand\n\\begin{equation}\\label{eq:B}\n\\big\\|D_{x_j}^{s+1\/2} u\\big\\|_{L^{\\infty}_{x_j}({\\mathbb R}; L^2_{x_1\\dots x_{j-1}x_{j+1}\\dots x_n,t}({\\mathbb R}^{n-1}\\times[0,T]))},\n\\end{equation}\n$j = 1, \\cdots ,n$, appearing in \\eqref{KPV}.\\\\\n\n{\\it Step 3.} To bound the quantity appearing in  \\eqref{eq:A}, first note that \\eqref{ouf} implies\n\\begin{equation*}\n\\big \\|\\sup_{0\\leq t\\leq T}\\quad\\sup_{x_1\\dots x_{j-1}x_{j+1}\\dots x_n }|e^{it\\Delta}u_0|\\;\\big \\|_{L^{4k}_{x_j}({\\mathbb R})}\n\\lesssim \\big\\|\\sup_{0\\leq t\\leq T} |e^{it\\Delta} J^s_{\\hat \\jmath} u_0|\\;\\big\\|_{L^{4k}({\\mathbb R}^n)},\n\\end{equation*}\nwith $s>\\frac{(n-1)}{4k}$. This estimate can be extended using Lemma \\ref{lem:B} by \nobserving that \n\\begin{equation*}\\label{une}\n\\begin{aligned}\n\\Big\\|\\sup_{0\\leq t\\leq T} |e^{it\\Delta} J^s_{\\hat \\jmath} u_0|\\;\\Big\\|_{L^{4k}({\\mathbb R}^n)} \n\\leq \\big\\| J^s_{\\hat \\jmath} u_0\\big\\|_{H^{\\sigma}({\\mathbb R}^n)}\\lesssim \\big\\|u_0\\big\\|_{H^{\\sigma+s}({\\mathbb R}^n)}=\\big\\|u_0\\big\\|_{H^{\\bar{s}}({\\mathbb R}^n)},\n\\end{aligned}\n\\end{equation*}\nwhere\n\\[\n\\bar{s} =s+\\sigma >\\frac{n-1}{4k}+n\\Big(\\frac{1}{2}-\\frac{1}{4k}\\Big)=\\frac{n}{2}-\\frac{1}{4k}.\n\\]\nInserting this inequality in the Duhamel representation \\eqref{eq:duhamel} with \n\\[\ns\\in\\Big(\\frac{n}{2}-\\frac{1}{4k},\\frac{n}{2}\\Big]=\\Big(\\frac{n}{2}-\\frac{1}{2p},\\frac{n}{2}\\Big],\n\\]\nit follows that\n\\begin{equation*}\n\\big\\|u\\big\\|_{L^{4k}_{x_j}({\\mathbb R};L^\\infty_{x_1\\cdot\\cdot x_{j-1}x_{j+1}\\cdot\\cdot x_n t}({\\mathbb R}^{n-1}\\times[0,T]))}\\leq C\\big\\|u_0\\big\\|_{s,2}+\n\\big\\|J^s(|u|^pu)\\big\\|_{L^1_t([0,T]; L^2({\\mathbb R}^n))}.\n\\end{equation*}\nSince $p=2k$, use of  a fractional Leibniz rule (see \\cite{KP}) implies that\n\\begin{equation}\\label{KaPo}\n\\big\\|J^s(|u|^{2k}u)\\big\\|_{L^1([0,T]; L^2({\\mathbb R}^n))}\\leq c\\big\\|u\\big\\|^{2k}_{L^{2k}([0,T];L^\\infty({\\mathbb R}^n))}\\big\\|J^su\\big\\|_{L^\\infty([0,T];L^2({\\mathbb R}^n))}.\n\\end{equation}\n\nIf it was known that \n\\begin{equation}\\label{enfin}\n\\|u\\|_{L^{2k}([0,T];L^\\infty({\\mathbb R}^n)))}\\leq cT^{\\theta}\\|J^su\\|_{L^q([0,T];L^r({\\mathbb R}^n))}\n\\end{equation}\nfor some $\\theta>0$ and for some admissible Strichartz pair $(r,q)$, then the sequence \nof inequalities could be closed. To obtain \\eqref{enfin},  recall that \n\\[\ns>\\frac{n}{2}-\\frac{1}{4k}=\\frac{2kn-1}{4k}=\\frac{n}{4kn\/(2kn-1)},\n\\]\nso we can \ntake \n\\[\nr=\\frac{4kn}{(2kn-1)}<\\frac{2n}{(n-2)}, \\quad \\text{if $n\\geq 3$}, \n\\]\n(the cases $n=1$ or $2$ are immediate) and $q=8k.$ By Sobolev embedding, the inequality  \\eqref{enfin} \nis seen to hold with $\\theta=\\frac{3}{8k}$. In summary, all the terms in  \n  \\eqref{eq:A} are shown to be bounded.\\\\\n\n{\\it Step 4.} Finally, attention is turned to terms of the form appearing in \\eqref{eq:B}. \n The local smoothing estimate enunciated in Lemma \\ref{lem:smooth} together \nwith  Duhamel's formula imply that  \n\\begin{eqnarray*}\n&\\big\\|D_{x_j}^{s+1\/2} u\\big\\|_{L^{\\infty}_{x_j}({\\mathbb R}; L^2_{x_1\\cdot \\cdot x_{j-1}x_{j+1}\\cdot \\cdot x_n,t}({\\mathbb R}^{n-1}\\times[0,T]))}\n\\\\\n&\\hspace{.8cm}   \\leq \\big\\|u_0\\big\\|_{H^{s}({\\mathbb R}^n)}+\\big\\|J^s(|u|^{2k}u)\\big\\|_{L^1([0,T];L^2({\\mathbb R}^n))}.\n\\end{eqnarray*}\nThe right-hand side was already estimated in \\eqref{KaPo}--\\eqref{enfin}. \nBecause of \\eqref{KPV}, this shows that  there exists a $C=C(T,n)>0$ such that\n\\[\n\\big\\|D_{x_j}^{s+1\/2} I\\big\\|_{L^\\infty([0,T]:L^2({\\mathbb R}^n))} \\leq  C.\n\\]\nSumming these estimates over $j$ for  $j=1\\cdots,n$  yields the result advertised in \nthe proposition.   \n\nFinally, we remark that in the case where  $p$ is not an even integer, the restriction \non $p$ are necessary and one needs \nto supplement  the Leibnitz-type inequality \\eqref{inequality} with\nthe chain rule for fractional derivatives adduced in the Appendix of \\cite{KPV93}. \n\\end{proof}\n\n\n \n \n\n\n\\subsection{An even stronger smoothing property}\n\nFor large $s$ and higher values of $p$, a stronger smoothing \n result than  that established in Proposition \\ref{prop:key} holds.\n\\begin{proposition}\\label{prop:keybis}\nLet $u_0\\in H^s({\\mathbb R}^n), s > \\frac{n}{2} - \\frac{1}{2p}$ with $p\\geq 2$ and \n$\\lfloor p+1\\rfloor \\geq s + \\frac{1}{2}$ if $p$ is not an even integer.  \nUnder these hypothses, it follows that for any $\\varepsilon > 0$,  \n\\[\nI\\in C([0,T];H^{s+1-\\varepsilon}({\\mathbb R}^n)),\n\\]\n where the  notation is taken from Proposition \\ref{prop:key} \n\\end{proposition}\n\\begin{remark}The loss of $\\varepsilon$ in the regularity of $I$ is needed to obtain a factor \n$T^{\\delta(\\varepsilon)},  \\delta(\\varepsilon)>0$, on the right-hand side of the inequalities below which \nallows them to be closed. \nIt can be recovered by assuming that the data $u_0$ is small enough in $H^s({\\mathbb R}^n).$\n\\end{remark}\nThe proof of Proposition {\\ref{prop:keybis} uses the following smoothing estimate, which is a direct consequence of \nLemma \\ref{lem:smooth}, a duality argument and the  Christ-Kiselev lemma \\cite{CK}. For a proof, see \\cite[Chapter 4]{LP}.\n\n\\begin{lemma}\\label{lem:Kis} For any $n\\in {\\mathbb N}$, the inequality\n\\[\n\\left\\|D^{1\/2}_{x_j}\\int_0^te^{i(t-s)\\Delta}f(\\cdot,s) \\, ds\\right\\|_{L^\\infty({\\mathbb R};L^2({\\mathbb R}^n))}\n\\leq C\\big\\|\\mathcal H_j f \\big\\|_{L^1_{x_j}({\\mathbb R};L^2_{x_1...x_{j-1}x_{j+1}...x_n t}({\\mathbb R}^n))}\n\\]\nholds, where $\\mathcal H_j$ denotes the {\\it Hilbert transform} in the $j$-th variable, which \nis to say, \n\\[\n\\mathcal H_j f(x):=-i \\, \\mathcal F^{-1} \\Big ( \\text{{\\rm sign}}(\\xi_j) \\widehat f(\\xi) \\Big)(x) .\n\\]\n\\end{lemma}\n\n\n\\begin{proof}[Proof of Proposition \\ref{prop:keybis}] \nThe proof is similar to that of Proposition \\ref{prop:key} and hence, we \nonly sketch the main differences. \n\nFirst consider the case of  data $u_0\\in H^s({\\mathbb R}^n)$\nwhich is small, so that all the norms involved are indeed \\lq\\lq small\". \nWe want to show that the integral term $I$ in Duhamel's formula is one order smoother in the Sobolev scale $C([0,T];H^s({\\mathbb R}^n))$ \nthan the the free propagation $e^{it\\Delta} u_0.$ To this end, apply Lemma \\ref{lem:Kis} together with the commutator estimate in \n\\cite[Theorem A.13]{KPV93} to write\n\\begin{equation}\\label{eq:L3}\n\\begin{aligned}\n&\\ \\big\\|\\mathcal H_j D^{s+1}_{x_j} I \\big\\|_{L^\\infty([0,T];L^2({\\mathbb R}^n))} \\\\ \n&\\hspace{.8cm} \\lesssim \n\\big\\| D^{s+1\/2}_{x_j}(|u|^{2k}u) \\big\\|_{L^1_{x_j}({\\mathbb R};L^2_{x_1\\cdot\\cdot x_{j-1}x_{j+1}\\cdot\\cdot x_n t}({\\mathbb R}^{n-1}\\times[0,T] ))}\\\\ \n& \\hspace{.8cm} \\lesssim\\Big (\\|u\\big\\|^{2k}_{L^{2k}_{x_j}({\\mathbb R};L^2_{x_1\\cdot\\cdot x_{j-1}x_{j+1}\\cdot\\cdot x_n t}({\\mathbb R}^{n-1}\\times[0,T] ))} \\\\\n&\\hspace{1.3cm} \\times \\|D_{x_j}^{s+1\/2}u\\|_{L^{\\infty}_{x_j}({\\mathbb R};L^2_{x_1\\cdot\\cdot x_{j-1}x_{j+1}\\cdot\\cdot x_n t}({\\mathbb R}^{n-1}\\times[0,T] ))}+R\\Big).\n\\end{aligned}\n\\end{equation}\nTo estimate the  two explicit quantities on the right-hand side of the last inequality,  \none uses arguments similar to those given in the proof of Proposition \\ref{prop:key}. \nThe estimates for the remainder terms represented by $R$ then follow by interpolation \nof the previous estimates. Since the terms on the right-hand side of \\eqref{eq:L3} are \nquadratic and each factor is small, one can close the estimate and get the desired result, but only provided that $u_0$ is sufficiently small.\n\nFor data $u_0\\in H^s({\\mathbb R}^n)$ of arbitrary size one gives up $\\varepsilon$-amount of spatial\nsmoothing for a little temporal smoothing, thereby obtaining the \n factor  $T^{\\delta(\\epsilon)}$, $\\delta(\\varepsilon)>0$  on the  right-hand side.  The \nright-hand side of the estimate then has lower homogeneity than the left side \nand the proof proceeds.\n\\end{proof}\n\n\n\n\\subsection{Extension to the case of non-elliptic Schr\\\"odinger equations} The results above extend to the case of non-elliptic, non-degenerate, nonlinear Schr\\\"odinger equations of the form\n\\begin{equation}\\label{eq:hypNLS}\ni\\partial_t u+\\Delta_{\\rm H} u\\pm |u|^p u=0,\\quad u\\big|_{t=0}=u_0(x),\n\\end{equation}\nwhere \n\\[\n\\Delta_{\\rm H} := \\partial^2_{x_1} + \\dots  \\partial^2_{x_j} -  \\partial^2_{x_{j+1}}\\dots -  \\partial^2_{x_n}.\n\\]\n\\begin{proposition}\\label{prop:hypNLS}\nThe result of Theorem \\ref{thm:main} also holds  for the initial-value \nproblem delineated in  \\eqref{eq:hypNLS} .\n\\end{proposition}\n\\begin{proof} Remark first that for initial data  of the form \n\\begin{equation}\n\\label{data-hyp}\n\\widetilde u_0(x)=\n\\frac{e^{-i\\alpha\\big((x_1-q_1)^2+\\dots+(x_j-q_j)^2-(x_{j+1}-q_{j+1})^2-\\dots-(x_n-q_n)^2\\big)}}{(1+|x|^2)^m},\n\\end{equation}\nwith $n\/4<m\\leq n\/2$, the solution of the  initial-value problem associated to \nthe linearization of \\eqref{eq:hypNLS} around the rest state, \ni.e., \\[u(x,t) =e^{it\\Delta_H}\\widetilde u_0(x),\\] \nsatisfies the conclusions of Lemma \\ref{lem:IC}, and in particular, blows up \n at the point $ \\widetilde q=(q_1,\\dots,q_j,-q_{j+1},\\dots,-q_n)$.\n\n\nNext,  notice  that the the local well-posedness result stated in Proposition \\ref{prop:CW}, is solely \nbased on Strichartz estimates, which are exactly the same for the two groups $\\{e^{it \\Delta} \\}_{t\\in {\\mathbb R}}$ and \n$\\{e^{it {\\Delta_{\\rm H}}} \\}_{t\\in {\\mathbb R}}$ ({\\it cf.} \\cite{GS}). \nIn other words, Lemma \\ref{lem:Strich} also holds in the  non-elliptic case.\n Together with Sobolev embeddings, this yields a unique solution $u\\in W_{s,n}^{T}$ to \\eqref{eq:hypNLS}, \nby the same arguments as in \\cite{CW}. Furthermore, the local smoothing estimates stated in Lemma \\ref{lem:smooth} \ncarry over to the  non-elliptic situation.   \n\nFinally, for the boundedness of the associated maximal function \n(see Lemma \\ref{lem:B}), we point to the \ninequality\n\\begin{equation}\n\\label{max-non-ellip}\n\\Big \\| \\sup_{0\\leq t\\leq T} \\big |e^{it(\\partial^2_{x_1} - \n\\partial^2_{x_2})}f \\big| \\,                              \\Big \\|_{L^4({\\mathbb R}^n)} \\leq C_T \\Big\\| D_x^{1\/2} f \\Big\\|_{L^2({\\mathbb R}^2)},\n\\end{equation}\nproved in \\cite[Theorem 2.6]{RVV} and observe that the same argument used there\n to establish \\eqref{max-non-ellip} shows that\n\\[\n\\Big \\| \\sup_{0\\leq t\\leq T}  \\big|e^{it \\Delta_{\\rm H} }f \\big| \\, \\Big\\|_{L^4({\\mathbb R}^n)} \\leq \nC_T \\Big\\| D_x^{n\/4} f  \\Big\\|_{L^2({\\mathbb R}^n)},\n\\]\nwhich is the desired estimate. The result then follows along the same lines as given in the proof of Theorem \\ref{thm:main}.\n\\end{proof}\n\nAn important consequence of this is the possibility of dispersive blow up for the Davey-Stewartson system in the \\lq\\lq elliptic\/elliptic\" or \\lq\\lq hyperbolic\/elliptic\" cases (see \\cite{GS} for more details \nabout  these systems, in particular for theory of local well-posedness).  This system arose originally \nas an approximate description of surface gravity-capillary waves in shallow water, but has \nother applications as well.   \n\n\\begin{corollary}\nFor  $\\alpha$ and $\\beta$ real and non-zero, consider the Davey-Stewartson system\n\\begin{equation}\n\\label{DS-ivp}\n\\left \\{ \n\\begin{aligned} \n&\\ i\\partial_t u\\pm \\partial^2_{x_1}u + \\partial^2_{x_2} u= \\alpha |u|^2 u + \\beta u \\partial_{x_1}\\phi ,\\;\\;\\;\\;\\;\\;\\;\\;\\;x=(x_1,x_2)\\in {\\mathbb R}^2, \\\\\n& \\ \\Delta \\phi = \\partial_{x_1} |u|^2.\n\\end{aligned}\n\\right.\n\\end{equation}\nThen there exist initial values $u_0\\in H^s({\\mathbb R}^2)\\cap L^{\\infty}({\\mathbb R}^2)\\cap C^{\\infty}({\\mathbb R}^2)$ with $s\\in (\\frac{1}{2},1]$ \nsuch that the solution $u$ of \\eqref{DS-ivp} with the initial condition $u(x,0)=u_0(x)$ exhibits dispersive blow up.\n\\end{corollary}\n\n\\begin{proof}\nRewrite the system \\eqref{DS-ivp} as the single equation\n\\begin{equation*}\ni\\partial_t u\\pm \\partial^2_{x_1}u + \\partial^2_{x_2} u= \\alpha |u|^2 u - \\beta u \\partial_{x_1}^2(-\\Delta)^{-1}(|u|^2),\n\\end{equation*}\nin the usual way.   The latter equation has the  equivalent form\n\\begin{equation*}\ni\\partial_t u\\pm \\partial^2_{x_1}u + \\partial^2_{x_2} u= \\alpha |u|^2 u - \\beta u R_1R_1 (|u|^2),\n\\end{equation*}\nwhere\n\\begin{equation*}\nR_1 f(x_1,x_2):= \\mathcal F^{-1}\\left(i\\frac{\\xi_1}{|\\xi|}\\widehat f(\\xi_1,\\xi_2)\\right)(x_1,x_2)\n\\end{equation*}\n denotes the $1$-Riesz transform in ${\\mathbb R}^2$\n\nThe $L^p$-continuity of the Riesz transform, \nimplies that the result in Proposition \\ref{prop:CW} and the argument entailed in the proof of\n Proposition \\ref{prop:key} still hold. Hence,  Theorem \n\\ref{thm:main} extends to solutions of the system \\eqref{DS-ivp}. \\end{proof}\n\n\n\n\n\n\\section{Dispersive blow up in the Gross-Pitaevskii  equation}\\label{sec:GP}\n\nIn this section, the discussion is moved to the initial-value problem \n\\begin{equation} \\label{eq:GP} \ni \\partial_t \\psi + \\Delta \\psi+ (1-|\\psi|^2)\\psi=0, \\quad \\psi\\big|_{t=0} = \\psi_0(x)\n\\end{equation}\nfor the Gross-Pitaevskii  equation.  \nHere $(x,t) \\in {\\mathbb R}^n\\times {\\mathbb R}$ and $\\psi$ is subject to the boundary condition \n\\begin{equation}\\label{eq:bc}\n\\lim_{|x|\\to \\infty} \\psi(x,t) = 1, \\quad \\text{for all $t\\in {\\mathbb R}$.}\n\\end{equation}\nThe Gross-Pitaevskii equation arises, for example, in the description of Bose-Einstein condensates, superfluid helium $\\text{He}^2$ and, \nin one spatial dimension, as a model for light propagation in a fiber \noptics cable (see for instance the survey article \\cite{BGS3} and other articles in the same volume). \n\\begin{remark} An important conserved quantity of \\eqref{eq:GP} is the Ginzburg-Landau energy, defined by\n\\[\nE(\\psi) = \\frac{1}{2} \\int_{{\\mathbb R}^n} |\\nabla \\psi|^2 \\, dx + \\frac{1}{4} \\int_{{\\mathbb R}^n} (1 - |\\psi|^2)^2 \\, dx.\n\\]\nThis invariant points to a natural energy space for the Gross-Pitaevskii equation, namely \n\\[\n\\mathcal{E}({\\mathbb R}^n) = \\{ \\psi \\in H^1_{\\text{loc}}({\\mathbb R}^n),  E(\\psi) < + \\infty \\}\n\\]\n (see \\cite{Ge1, Ge2} for more details).\n\\end{remark}\n\n\n\n\n\n\\subsection{A reformulation of the Gross-Pitaevskii Equation}\n\nDue to the non-zero boundary condition \\eqref{eq:bc} at infinity, the \nideas that worked in the earlier sections do not apply directly to show \ndispersive blow up.    \nTo prove dispersive blow up for the Gross-Pitaevskii equation, make the change \n \\[\\psi(t,x) =1+u(t,x),\\] \nof the dependent variable so that $u\\in L^2({\\mathbb R}^n)$ describes the deviation from the steady state. In terms of $u$, the \nGross-Pitaevskii equation \\eqref{eq:GP} becomes\n\\begin{equation}\\label{bife}\ni\\partial_t u +\\Delta u-2 \\mathrm{Re} \\,u=F(u) , \\quad  u \\big|_{t=0} = \\psi_0(x)-1,\n\\end{equation} \nwhere \n\\[\nF(u) = u^2+2|u|^2+|u|^2u.\n\\]\nEven for this  transformed initial-value problem \\eqref{bife}, dispersive blow up\n does not fall directly to the general lines of \nargument proposed earlier.  This is due to the appearance of \nthe slightly odd, ${\\mathbb R}$-linear term $-2 \\mathrm{Re}\\,  u$, which at least\n in principle might cancel out the effect of dispersive blow up stemming from the Laplacian. \n\nThis obstacle will be surmounted by using the further reformulation  developed \nby Gustafson, Nakanishi and Tsai in \\cite{GNT1, GNT2}. \nFollowing their lead,  introduce the\nFourier multipliers\n\\begin{equation}  \\label{operators}\nA:=  \\sqrt{-\\Delta(2-\\Delta)} \\quad {\\rm and} \\quad  B:= \\sqrt{-\\Delta(2-\\Delta)^{-1}}\\, .\n\\end{equation}\nThese operators satisfy $A = -\\Delta B^{-1} = (2- \\Delta)B=\\sqrt{-\\Delta(2-\\Delta)}$. Next,  define the ${\\mathbb R}$-linear operator\n\\[\n\\Upsilon u := B\\, \\mathrm{Re}\\, u + i \\,\\mathrm{Im} \\, u\\equiv B u_1 + i u_2, \n\\]\nwhere $u = u_1 + iu_2$.   \nOne checks (see \\cite{GNT1, GNT2}) that the left-hand side of the equation in \\eqref{bife} \ncan be written in the form\n\\begin{equation}\\label{eq:identity}\ni\\partial_t u +\\Delta u-2 \\mathrm{Re} \\,u =  \\Upsilon (i\\partial_t  - A ) \\Upsilon^{-1}u.\n\\end{equation}\nDenote by $v$ the function\n\\[\nv:=\\Upsilon^{-1}u \\equiv \\Upsilon^{-1}(u_1+iu_2)=B^{-1}u_1+iu_2\n\\] \nand rewrite \\eqref{bife} as\n\\begin{equation}\\label{eq:reGP}\ni \\partial_t v - A v = \\Upsilon^{-1}F(u), \\quad v\\big|_{t=0} = \\Upsilon^{-1} (\\psi_0-1).\n\\end{equation}\nThe associated free evolution is \n\\[\nw(x,t)=e^{-itA} v_0 (x) := \\frac{1}{(2\\pi)^n} \\int_{{\\mathbb R}^n} e^{it\\sqrt{|\\xi|^2(|\\xi|^2+2)}} \\widehat u_0(\\xi) e^{i \\xi\\cdot x} \\, d\\xi,\n\\]\nwhich can be represented as follows.\n\\begin{lemma}\\label{lem:decomp}\nFor any $f\\in L^2({\\mathbb R}^n)$ and any $t\\not =0$, the group $\\{e^{-itA}\\}_{t\\in {\\mathbb R}}$ has the representation\n\\[e^{-itA} f(x) = (G(\\cdot , t) \\ast f)(x),\\] where the kernel is $G =  G_1 + G_2$ with\n\\[\nG_1(x,t)   =\\frac{e^{it}}{(4\\pi it)^{n\/2}}\\, e^{\\frac{i|x|^2}{4t}}\\quad and \\quad G_2(x,t) := \\frac{e^{it} }{(2\\pi)^n} \\int_{{\\mathbb R}^n} \\kappa(\\xi, t) e^{it |\\xi|^2} e^{ix\\cdot \\xi}d\\xi.\n\\]\nHere, the convolution is with respect to the spatial variable over ${\\mathbb R}^n$ \nand the kernel  $\\kappa $ is \n\\[\n\\kappa(\\xi, t)=2i\\sin \\Big(\\frac{t}{2}r(\\xi)\\Big)e^{i\\frac{t}{2} r(\\xi)}\n\\]\nwith \n\\[\nr(\\xi)= \\frac{-2|\\xi|^2}{\\left(\\sqrt{|\\xi|^2(|\\xi|^2+2)}+|\\xi|^2\\right)^2}\\sim O(|\\xi|^{-2}), \\ \\text{as $|\\xi|\\to \\infty$.}\n\\]\nMoreover, $r$ lies in $C_{\\rm b}({\\mathbb R}^n)$ and is smooth away from the origin.\n\\end{lemma}\n\nNote that $G_1$ is the usual Schr\\\"odinger group multiplied by $e^{it}$.  Remark also that, as $|\\xi|\\to \\infty$, $\\kappa(\\cdot, t)$ decays to zero like $|\\xi| ^{-2}$, \nuniformly on compact time-intervals. \n\n\\begin{proof} \nThe convolution kernel    $G$ is\n\\[G(x,t)=\\mathcal F^{-1}\\left(e^{it \\sqrt{(|\\xi|^2(|\\xi|^2+2)}}\\right)(x,t).\\]\nObserve that \n\\[\n\\sqrt{(|\\xi|^2(|\\xi|^2+2)}=|\\xi|^2+a(\\xi),\n\\]\nwhere \n\\[\na(\\xi)=\\frac{2|\\xi|^2}{\\sqrt{(|\\xi|^2(|\\xi|^2+2)}+|\\xi|^2}=1-\\frac{2|\\xi|^2}{\\left(\\sqrt{(|\\xi|^2(|\\xi|^2+2)}+|\\xi|^2\\right)^2}=1+r(\\xi).\\]\n Consequently, it follows that\n \\[\n e^{it \\sqrt{(|\\xi|^2(|\\xi|^2+2)}}=e^{it|\\xi|^2}e^{it}e^{itr(\\xi)}=e^{it}e^{it|\\xi|^2}(1+f_t(|\\xi|)),\n \\]\nwhere\n\\[\nf_t(|\\xi|)=2i\\sin \\Big(\\frac{t}{2}r(\\xi)\\Big)e^{it\\frac {r(\\xi)}{2}}\\]\nis continuous, smooth on ${\\mathbb R}^n$ and\ndecays to zero like $|\\xi| ^{-2}$, as $|\\xi|\\to \\infty$, uniformly on compact time intervals in $(0,\\infty)$,  since\n$r({|\\xi|})$ does so.\nThis allows  the propagator in Fourier space to be written as\n\\[\n\\widehat G(\\xi, t)= e^{it\\sqrt{|\\xi|^2(|\\xi|^2+2)}}=e^{it}e^{it|\\xi|^2}(1+\\kappa(\\xi, t))\n\\]\nwith $\\kappa$ as above. \n\\end{proof}\n\nWith this representation in hand, dispersive blow up for the evolutionary group $\\{e^{-itA}\\}_{t\\in {\\mathbb R}}$ \ncan be established for at least  spatial dimensions less than or equal to three. \n\n\\begin{lemma}\\label{lem:DBUv}\nLet $n\\leq 3$  and suppose given $x_*\\in {\\mathbb R}^n$, $t_*>0$ and $s \\in [0,\\frac{n}{2})$.  Then \nthere exist initial data $v_0\\in C^\\infty({\\mathbb R}^n)\\cap H^s({\\mathbb R}^n)\\cap L^\\infty({\\mathbb R}^n)$ such that\n\\[\nw(\\cdot, t) = e^{-itA} v_0 \\in H^s({\\mathbb R}^n)\n\\]\nexhibits dispersive blow up at $(x_*, t_*)$.\n\\end{lemma}\n\\begin{proof} Choose $(x_*,t_*) = (0,\\frac{1}{4})$ without loss of generality. \n\tAs in Lemma \\ref{lem:linear}, let \n\\begin{equation} \\label{initdata}\nv_0(x)=\\frac{e^{-i|x|^2}}{(1+|x|^2)^m}, \\quad \\text{with  $\\frac{n}{4}<m\\leq \\frac{n}{2}$.}\n\\end{equation}\nUsing Lemma \\ref{lem:decomp},  it is found that \n\\[\ne^{-itA} v_0(x) = (G_1(\\cdot , t) \\ast v_0)(x) +(G_2(\\cdot , t) \\ast v_0)(x).\n\\]\nIn view of the calculations in Section \\ref{sec:lin} it is inferred \nthat the first term on the right-hand side will exhibit dispersive blow up at $(0,\\frac{1}{4})$.\nIt thus suffices to show that  $G_2(\\cdot,t)\\ast v_0\\in C_{\\rm b}( {\\mathbb R}^n)$, uniformly on compact time-intervals. \n\nThe kernel $G_2(\\cdot ,t)\\in L^2({\\mathbb R}^n)$ for $n\\leq 3$, due to the decay properties of $r(\\xi)$. Since $v_0\\in L^2({\\mathbb R}^n)$ by \nconstruction,  the product $\\widehat G_2 (\\cdot , t) \\widehat v_0\\in L^1({\\mathbb R}^n)$, uniformly on compact time-intervals.  The Riemann-Lebesgue lemma then implies the desired result.\n\\end{proof}\n\nNote that even though $e^{-it A}$ represents the linear evolution operator associated to \\eqref{eq:reGP}, it includes part of the nonlinearity of the \noriginal Gross-Pitaevskii equation \\eqref{eq:GP}, as can be seen from \\eqref{eq:identity}.\n\n\n\n\n\\subsection{Proof of dispersive blow up for Gross-Pitaevskii} \n\n\nLemma \\ref{lem:DBUv} together with the results of Section \\ref{sec:NLS} now allow us to prove \ndispersive blow up for equation \\eqref{eq:reGP} and, consequently, also for the original \nGross-Pitaevskii equation (in physically relevant dimensions $n = 1,2,3$). \nTo effect a proof, the following lemma on the  mapping properties of $B$ and its inverse will be used.\n\\begin{lemma}\\label{mapping}\nThe operator $B$ in \\eqref{operators} may be written in the form\n\\[B={\\rm Id}+B_1, \\quad B^{-1}={\\rm Id} +B_2,\\] \nwhere $B_1$, $B_2\\in \\mathcal L(H^s({\\mathbb R}^n),H^{s+2}({\\mathbb R}^n))$, for all $s\\in {\\mathbb R}.$ \n\\end{lemma}\n\n\\begin{proof}\nThe proof is a simple computation on the level of the Fourier symbols $b_1$ and $b_2$ associated to $B_1$ and $B_2.$ For example, the symbol $b_1$ is\n\\[\nb_1(\\xi)=-\\frac{2}{(2+|\\xi|^2)( \\sqrt{|\\xi|^2(2+|\\xi|^2)^{-1}}+1)}.\n\\]\nFrom this formula, one sees that $b_1$ is  in fact uniformly bounded and \nis $O(|\\xi|^{-2})$ as $|\\xi| \\to \\infty$. Similarly, one finds\n\\[b_2(\\xi)=\\frac{2}{(2+|\\xi|^2)\\sqrt{|\\xi|^2(2+|\\xi|^2)^{-1}}[1+(|\\xi|^2(2+|\\xi|^2)^{-1}]^{1\/2}},\\]\nfrom which one infers the assertion about $B_2$.\n\\end{proof}\n\n\nIn particular, since $\\Upsilon^{-1}u=B^{-1}u_1+iu_2$, the last lemma\n implies that, for all $s\\in {\\mathbb R}, \\Upsilon$ and  $\\Upsilon ^{-1}$ both lie in \n$\\mathcal L(H^s({\\mathbb R}^n),H^{s}({\\mathbb R}^n))$. \n\nHere is  the main result of this section.\n\n\\begin{theorem} \\label{DBU:v}\nLet $n\\leq 3$, $x_*\\in {\\mathbb R}^n$, $t_*>0$ and $s \\in (0,\\frac{n}{2})$ be given.  Then\nthere exist initial data $v_0\\in C^\\infty({\\mathbb R}^n)\\cap H^s({\\mathbb R}^n)\\cap L^\\infty({\\mathbb R}^n)$ \nsuch that the corresponding solution $v\\in C([0,T], H^s({\\mathbb R}^n))$ of \\eqref{eq:reGP} exhibits dispersive blow up at $(x_*,t_*).$\n\n\\end{theorem}\n\\begin{proof} Take $v_0$ in the form  \\eqref{initdata} in the proof of Lemma \\ref{lem:DBUv}. \nBy an approprite choice of $m$, Lemma \\ref{lem:linear},  guarantees that $v_0$ lies \nin $H^s({\\mathbb R}^n)$.  \nNext, in view of Lemma \\ref{mapping}, the corresponding $u_0=\\Upsilon^{-1} v_0\\equiv B^{-1}\\text{Re} \\, v_0 + i \\text{Im} \\, v_0$ has the same Sobolev regularity as $v_0$. \nFor such a $u_0$, the Cazenave-Weissler theory can be applied to the Gross-Pitaevskii equation \\eqref{bife}, yielding a local in-time solution $u\\in C([0,T], H^s({\\mathbb R}^n))$ for the specified\n $s<\\frac{n}{2}$, which  \n leads to a solution $v\\equiv \\Upsilon u \\in C([0,T], H^s({\\mathbb R}^n))$ of \\eqref{eq:reGP}. \nConsequently,  we can use the strategy followed for the usual \nnonlinear Schr\\\"odinger equation. \nTo this end, write \\begin{equation}\\label{eq:duhamelv}\nv(x,t)=e^{-it A}v_0 (x)+ i \\int_0^te^{-i(t-s)A}  \\Upsilon^{-1}F(u(x,s)) \\, ds,\n\\end{equation}\nwhere the nonlinearity is explicitly given by \n\\[\n \\Upsilon^{-1}F(u) =  B^{-1} (3 u_1^2+u_2^2+|u|^2u_1)+iu_2(2u_1+|u|^2),\n\\]\nsince $u=u_1+iu_2$. As before, the first term on the right-hand side exhibits dispersive blow up at $(x_*, t_*)=(0,\\frac{1}{4})$ provided $n<4$. \nThe advertised result will be in hand when the second term is known to be uniformly bounded.\n\nThe integral term in \\eqref {eq:duhamelv} splits into $I_1+I_2$ where\n\\[\nI_j(x,t)=\\int_0^t G_j(\\cdot, t-s)\\ast \\Upsilon^{-1}F(u(\\cdot,s)) \\, ds,\\quad j=1,2,\n\\]\ncorresponding to $G=G_1+G_2$.\nThe fact that $I_2$ is a bounded continuous function can be concluded by the same argument \nthat prevailed in the proof \nof Lemma \\ref{lem:DBUv}. For $n<4$, $G_2(\\cdot, t)\\in L^2({\\mathbb R}^n)$, and so is $\\Upsilon^{-1} F(u)$ \nbecause\n $u\\in H^s({\\mathbb R}^n)$ and $\\Upsilon^{-1}\\in \\mathcal L(H^s({\\mathbb R}^n),H^{s}({\\mathbb R}^n))$.  \n\nOn the other hand, $I_1$ is given by\n\\[\n I_1(x,t)=\\int_0^t G_1(\\cdot, t-s)\\ast \\big( B^{-1}[3u_1^2+u_2^2+|u|^2u_1]+u_2(2u_1+|u|^2)\\big) (\\cdot,s) \\, ds.\n \\]\nInasmuch as  $G_1$ is, up to a multiplicative constant, the fundamental solution of the usual linear Schr\\\"{o}dinger equation \n(see Lemma \\ref{lem:decomp}), the double  integral $I_1$ \nis of the same form (up the the appearance of $B^{-1}$) as the usual nonlinearity in the Duhamel representation of the nonlinear Schr\\\"odinger equation.\nIn view of the mapping properties of $B^{-1}$, \nthe desired result of boundedness and continuity \nthen follows from the corresponding proof for the usual nonlinear Schr\\\"odinger  equation \ngiven in Sections \\ref{sec:DBUNLS} and \\ref{sec:global}.\n\\end{proof}\n\n\n\n\n\n\n\nAs a corollary, we infer the appearance of dispersive blow up for the original \nGross-Pitaevskii equation \\eqref{eq:GP} in physically relevant dimensions.\n\n\n\\begin{corollary} Let $n\\leq 3$. Given $x_*\\in {\\mathbb R}^n$, $t_*>0$, there exist smooth and bounded initial data $\\psi_0 \\in C_{\\rm b}({\\mathbb R}^n)$ with \n$\\psi_0-1\\in L^2({\\mathbb R}^n)$, such that the solution $\\psi$ exhibits \ndispersive blow up at $(x_*, t_*)$.\n\\end{corollary}\n\n\n\\begin{proof}  To establish dispersive blow up for \\eqref{eq:GP},  the results for $v$ \nneed to be transferred back to the\nvariable $\\psi$.   In search of such a conclusion, note that the initial data in \\eqref{eq:reGP} \nis given by\n\\[\nv_0(x) = \\Upsilon^{-1} (\\psi_0-1) = B^{-1}(\\mathrm{Re} \\, \\psi_0 -1) + i \\, \\mathrm{Im} \\, \\psi_0.\n\\]\nThe specific choice $v_0(x)= \\frac{e^{-i|x|^2}}{(1+|x|^2)^m}$ then corresponds to the  initial data \n\\[\n\\psi_0 (x) = 1+ B \\, \\mathrm{Re} \\, v_0(x)  + i \\, \\mathrm{Im} \\, v_0(x) = 1+ B \\left(\\frac{\\cos |x|^2}{(1+|x|^2)^m} \\right) +  \\frac{i \\sin |x|^2}{(1+|x|^2)^m} \n\\]\nfor \\eqref{eq:GP}.  Since $B:L^2({\\mathbb R}^n)\\to L^2({\\mathbb R}^n)$, it is clear from the last formula \nthat $\\psi_0 - 1 \\in L^2({\\mathbb R}^n)$.  By Lemma \\ref{mapping}, $B={\\rm Id}+ B_1$ with  $B_1\\in \\mathcal L(L^2({\\mathbb R}^n), H^2({\\mathbb R}^n)) $ and $H^2({\\mathbb R}^n)\\subset C_{\\rm b}({\\mathbb R}^n)$ for $n\\leq 3,$ so \nit is concluded that $\\psi_0\\in C_{\\rm b}({\\mathbb R}^n).$\n\nNow, if $v=v_1+iv_2$ is a solution of  \\eqref{eq:reGP} with the dispersive blow up \nproperty at a point $(x_*,t_*)$ provided by Theorem \\ref{DBU:v}, \nthe corresponding $u =u_1+iu_2$ is given by\n\\[u_1=B^{-1}v_1=(I+B_2)v_1,\\quad u_2=v_2.\\]\nSince $B_2$ is a smoothing operator, $u,$  and thus $\\psi=1+u$ satisfies the DBU property at the same point $(x_*,t_*).$  \n\\end{proof}\n\n\n\n\\section{Higher-order nonlinear Schr\\\"odinger equations} \\label{sec:higherNLS}\n\nIn this final section, we indicate how  results of dispersive blow up can be extended \nto higher-order nonlinear Schr\\\"odinger equations.  It is  mathematically \nnatural to inquire whether or not higher-order terms destroy dispersive blow up, \nbut the practical motivation for considering such an extension is perhaps even more \ntelling.   In nonlinear optics, third and fourth-order Schr\\\"odinger-type \nequations frequently appear in the description of various  wave phenomena. \nIn particular, the analysis of optical rogue-wave formation has been based on higher-order \nnonlinear Schr\\\"odinger equations  (see, for example, \\cite{D, DGE, Mu, T1}).\n\nAs the ideas and even much of the technical detail mirror closely what has gone\nbefore, we content ourselves with admittedly sketchy indications of how the theory\nis developed.   The one point which would require serious new effort has to do \nwith an appropriate generalization of the Cazenave--Weissler theory in \\cite{CW} \nto a higher-order setting.  This is not attempted here, but is deserving of \nfurther investigation at a later stage.   \n\n\n\n\n\\subsection{Fourth-order nonlinear Schr\\\"odinger equation}\nIn this subsection,  initial-value problems for \nfourth-order nonlinear  Schr\\\"{o}dinger equations of the form \n\\begin{equation}\\label{eq:4th}\ni\\partial_t u +\\alpha \\Delta u+\\beta \\Delta ^2 u+\\lambda |u|^pu=0, \\quad u\\big|_{t=0}=u_0(x),\n\\end{equation}\n are considered.  Here, the parameters  $\\alpha, \\beta, \\lambda$ are real constants, \n with $\\beta \\neq 0$.  \nTheory for this initial-value problem can be found, for example, in \\cite{FES, P} and in \nthe references cited in these works.   \nIf $\\alpha =0$, the partial differential equation\n is often referred to as the  bi-harmonic NLS equation  (see, e.g., \\cite{BFM}). \n A simple scaling allows us to assume $\\beta=1$ and to consider only the values \n$\\alpha \\in  \\lbrace 0, -1, +1\\rbrace$, though time may need to be reversed.\n\nTo establish dispersive blow up for \\eqref{eq:4th},  the dispersive properties \nof the associated linear equation\n\\begin{equation}\\label{eq:lin4th}\ni \\partial_t u -\\alpha  \\Delta u+ \\Delta^2 u=0, \\quad \\alpha \\in \\lbrace 0, -1, +1\\rbrace\n\\end{equation}\nare helpful, just as for the lower-order cases.  As should be  clear from the preceding theory, \n the possibility of dispersive blow up for \\eqref{eq:lin4th} is  linked to \nthe dispersive  properties of the fundamental solution \n\\begin{equation}\\label{4disp}\n\\Sigma_\\alpha (x,t)= \\frac{1}{(2\\pi)^{n\/2} } \\int_{{\\mathbb R}^n} e^{it(|\\xi|^4+\\alpha |\\xi|^2)+ix\\cdot \\xi} \\, dx\n\\end{equation}\nof \\eqref{eq:lin4th}, which have in fact been established already in \\cite{BAKS}.\n\n\\begin{lemma}\\label{4dec}\nLet $\\Sigma_\\alpha$ be as in \\eqref{4disp} and $\\mu \\in {\\mathbb N}^n$a multi-index. \n\\begin{enumerate}\n\\item If $\\alpha  =0$, there exists a $C>0$ such that for $x \\in {\\mathbb R}^n$ and $t> 0$,\n$$\n|\\partial^\\mu \\Sigma_0(x,t)|\\leq Ct^{-(n+|\\mu|)\/4}\\left(1+\\frac{|x|}{t^{1\/4}}\\right)^{-(n-|\\mu|)\/3}.\n$$\n\\item \nFor $t > 0$ and  either $t\\leq 1$ or $|x|\\geq t$,  there exists a $C>0$ such that\n$$\n|\\partial^\\mu \\Sigma_\\alpha(x,t)|\\leq \nCt^{-(n+|\\mu|)\/4}\\left(1+\\frac{|x|}{t^{1\/4}}\\right)^{-(n-|\\mu|)\/3}\n$$\nfor $\\alpha = \\pm 1$.\n\\end{enumerate}\n\\end{lemma}\n\nStrichartz estimates then follow pretty much directly from Lemma \\ref{4dec}. \n In some detail, we say that \nthe pair $(q,r)$ is admissible for the fourth-order Schr\\\"odinger group $\\{e^{it (\\Delta^2-\\alpha \\Delta)}\\}_{t\\in {\\mathbb R}}$ if\n\\begin{equation}\\label{eq:adm}\n\\frac{1}{q}=\\frac{n}{4}\\left(\\frac{1}{2}-\\frac{1}{r}\\right),\n\\end{equation}\nfor $2\\leq r\\leq \\frac{2n}{n-2}$ if $n \\geq 3$, respectively, $2\\leq r\\leq \\infty$ if $n=1$ \nand  $2\\leq r<\\infty$ if $n=2$. \nUsing this, one has the following estimates, which are the fourth-order counterpart to the ones \nreported in Lemma \\ref{lem:Strich}. In what follows, $T=+\\infty$ when \n$\\alpha =0$ and $T$ is any nonnegative number when $\\alpha= \\pm 1.$\n \n\\begin{lemma}[\\cite{BAKS}]\n\\label{Str}\nLet $(q, r)$ be admissible in the sense of \\eqref{eq:adm}. Then there exists $c=c(n, r,T)$ such that\n\\[  \\big \\|  e^{it (\\Delta^2-\\alpha \\Delta)} f \\big \\|_{L^q((-T,T);L^r({\\mathbb R}^n))}\\leq c\\|f \\|_{L^2({\\mathbb R}^n)}. \\]\nThe linear operator \n\\[\n\\Phi f= \\int_0^t e^{i(t-s)(\\Delta^2-\\alpha \\Delta)}f(s)ds\n\\]\nis bounded in the sense that \n\\[ \\| \\Phi f\\|_{L^q((-T,T);L^r({\\mathbb R}^n))}\\leq c\\|f\\|_{L^{q'}((-T,T);L^{r'}({\\mathbb R}^n))}, \\]\nwhere $\\frac{1}{q}+\\frac{1}{q'}=1$ and  $\\frac{1}{r}+\\frac{1}{r'}=1.$ \n\\end{lemma}\n\nIf we assume for the moment that dispersive blow up holds true for the linear model \\eqref{eq:lin4th}, then the \nStrichartz estimates above are already sufficient to prove dispersive blow up for the nonlinear equation \\eqref{eq:4th} in the physically relevant dimensions $n\\leq 3$. \n\n\n\\begin{proposition}\nLet $n\\leq 3$ and $\\alpha \\in \\lbrace 0, -1, +1\\rbrace$. Assume that the linear fourth-order equation \\eqref{eq:lin4th} exhibits dispersive blow up \nat some  point $(x_*, t_*)$ in space-time. Then, for $p < \\frac{8}{n}-1$, so does the fourth-order \ninitial-value problem \\eqref{eq:4th}.\n\\end{proposition}\n\n\\begin{proof}\nThe proof follows closely the one given in \\cite{BS2} for the one-dimensional, second-order   nonlinear Schr\\\"{o}dinger equation. \nIn particular, the fact that the dispersive estimate in Lemma \\ref{4dec} of the fundamental solution $\\Sigma_\\alpha$ has temporal behavior that goes like  $t\\mapsto t^{-n\/4}$, which is locally integrable for $n<4$, is a key point in the proof. \n\nThe Duhamel representation of \\eqref{eq:4th} is given by\n\\begin{align*}\nu(x,t) = & \\, e^{i t (\\Delta^2-\\alpha \\Delta)} u_0(x) + i \\lambda \\int_0^t e^{i(t-s)(\\Delta^2-\\alpha \\Delta)} |u(x,s)|^p u(x,s) \\, ds \\\\\n= & \\,  e^{i t (\\Delta^2-\\alpha \\Delta)} u_0(x) + i \\lambda I (x,t).\n\\end{align*}\nThe first term on the right-hand side exhibits dispersive blow up by assumption.  \nTo prove that the integral term \nis continuous and bounded, notice that  \n\\[\n|I(x,t) |\\leq  C\\int_0^t\\int_{{\\mathbb R}^n} \\frac{1}{(t-s)^{n\/4}}|u|^{p+1}(x-y,t-s)\\, ds\\, dy \n\\]\nusing Lemma \\ref{4dec}. Applying H\\\"{o}lder's inequality, with a $\\gamma\\in(0,4\/n)$ to be determined presently, it is found that\n\\[\n|I(t,x) | \\leq\\left(\\int_{\\mathbb R} \\frac{ds}{(t-s)^{n\\gamma\/4}}\\right)^{1\/\\gamma}\\left(\\int_{\\mathbb R}\\|u(\\cdot,s)\\|_{p+1}^{\\gamma'(p+1)} ds\\right)^{1\/\\gamma'},\n\\]\nwith $\\frac{1}{\\gamma}+\\frac{1}{\\gamma'}=1$. \nChoose an admissible Strichartz pair in the range \\eqref{eq:adm} as follows.\nTake $r=p+1$ so that $q=\\frac{8(p+1)}{n(p-1)}.$ The condition $\\gamma'(p+1)\\leq q$ then yields\n\\[\\gamma'=\\frac{\\gamma}{\\gamma -1} \\leq  \\frac{8}{n(p-1)}.\\] \nCombined with the condition $\\gamma <\\frac{4}{n}$, one obtains\n\\[\np < \\frac{8}{n} \\left( 1 - \\frac{1}{\\gamma}\\right) = \\frac{8}{n} -1,\n\\]\nand the assertion is proved.\n\\end{proof}\n\n\\begin{remark}\nThe strategy deployed in this proof does not yield the optimal range of exponents $p$ nor is \nit valid for $n \\geq 4$.  This is because \nit is based only on Strichartz estimates \nand because $t\\mapsto t^{-n\/4}$ is locally integrable only  for $n<4$. \nTo extend the proof to higher dimensions $n > 3$, and to more general \nnonlinearities $p>0$, one could argue as in the proof of Proposition \\ref{prop:key}.  However,\nan essential ingredient in our argument was the Cazenave-Weissler result \nrecounted in  Proposition \\ref{prop:CW}.\nThus, to carry out this line of reasoning successfully, we would need \n the analog of the Cazenave-Weissler results  in the case of fourth-order equations, as well as  \nthe corresponding smoothing estimates for the Duhamel term established in Section \\ref{sec:global}. These tasks will be the goal of an upcoming work.\n\\end{remark}\n\nTo close the analysis, it is still required to prove that the linear fourth-order equation \\eqref{eq:lin4th} does exhibit dispersive blow up. \nTo this end, we will need a more precise description of the decay of the fundamental solution. \nThis will only be carried out in the one-dimensional case,\n though the result in higher spatial dimensions does not require new ideas, \njust more complex calculations.\n\n\\begin{proposition}\\label{dbu4th}\nLet $n=1$. Given $(x^*,t^*)\\in {\\mathbb R}\\times ({\\mathbb R}\\setminus \\lbrace 0\\rbrace)$, there exists $u_0 \\in C^\\infty({\\mathbb R})\\cap L^2({\\mathbb R})\\cap L^\\infty({\\mathbb R})$ such that the \nsolution $u \\in C_{\\rm b}({\\mathbb R}; L^2({\\mathbb R}))$ to \\eqref{eq:lin4th} has the following properties.\n\\begin{enumerate}\n\\item The solution $u$ blows up at $(x_*,t_*)$ which is to say\n\\[\n\\lim _{(x,t)\\in {\\mathbb R}\\to (x_*, t_*)} | u(x,t)| = + \\infty.\n\\]\n\\item The function $u$ is continuous on $\\big\\{(x,t)$ on ${\\mathbb R}\\times {\\mathbb R} \\setminus \\{t_*\\}\\big\\}$.\n\\item  The solution $u(\\cdot, t_*)$ is a continuous function on ${\\mathbb R} \\setminus \\{x_*\\}$.\n\\end{enumerate} \n\\end{proposition}\n\n\\begin{proof} For $n=1$, it is readily checked that the fundamental solution of \n\\begin{equation}\\label{4th1D}\ni \\partial_t u_t-\\alpha \\partial_x^2 u + \\partial_x^4u =0\n\\end{equation}\nis given by\n\\[\n\\Sigma_\\alpha(x,t)=\\frac{1}{(4t)^{1\/4}}B\\left(2\\alpha t^{1\/2},\\frac{x}{(4t)^{1\/4}}\\right)\n\\]\nwhere  \n\\begin{equation*}\\label{pear}\nB(x,y)=\\frac{1}{\\pi} \\int_{\\mathbb R} e^{i\\big(\\frac{1}{4}s^4+\\frac{1}{2}xs^2+ys\\big)}\\, ds\n\\end{equation*}\n is the  Pearcey integral. The Pearcey integral is a smooth and bounded function of $(x,y) \\in {\\mathbb R}^2$ which decays in both variables thusly: \n\\begin{equation*}\\label{asPe}\n|B(x,y)|\\leq c\\big(1+y^2+|x|^3\\big)^{-1\/18}\\Big(1+(1+y^2+|x|^3)^{-5\/9}|(3y)^2+(2x)^3|\\Big)^{-1\/4}\n\\end{equation*}\n(see, e.g., \\cite{BAKS}).\nTo establish dispersive blow up for \\eqref{4th1D}, an asymptotic expansion of the Pearcey integral \nwith respect to the second variable $y$ is helpful. Such an expansion has been established in \\cite{Pa}, Formulas (2.14) and (5.12).\n\\footnote{Note the rotation of coordinates in those formulas as compared to the way they \nare written here.} These results imply that\n\\begin{itemize}\n\\item[(i)] in the case $\\alpha =0$, one has\n\\begin{equation*}\\label{Pe0}\nB(0,y)=C_1y^{-1\/3}+C_2 y^{-5\/3}+O(|y|^{-3}),\n\\end{equation*}\n as $|y|\\to +\\infty$, uniformly for bounded values of $x$, where $C_1$ and $C_2$ are nonzero complex constants and\n\\item[(ii)] when $\\alpha \\neq0$, \n\\begin{align*}\\label{Pe2}\nB(x,y)=2^{1\/6}e^{-i\\pi\/24}y^{-1\/3}e^{-i x^2\/6}\n\\exp\\Big(-\\frac{3i}{4^{4\/3}}y^{4\/3}+\\frac{i}{4^{2\/3}}xy^{-2\/3}]\\Big)\\times \\\\\n\\left(1+\\frac{4^{-1\/3}}{3}xy^{-2\/3}(1-\\frac{i}{9}x^2)+O(|y|^{-4\/3})\\right)\n\\end{align*}\nfor $|y|\\to +\\infty$ and bounded $x$.\n\\end{itemize}\n   \n\nWith these results in hand, an argument can be mounted that mimics\n the case of the usual Schr\\\"odinger group. Without loss of generality, take it \nthat  $(x_*,t_*) =(0,\\frac{1}{4})$ and consider in \\eqref{4th1D} the initial data \n\\[\nu_0(x)=\\frac{\\Sigma_0(-x,\\frac{1}{4})}{(1+x^2)^m}=\\frac{B(0,-x)}{(1+x^2)^m},\\quad \\frac{1}{12}<m\\leq \\frac{1}{6},\n\\]\nif $\\alpha =0,$ and\n\\[\nu_0(x)=\\frac{\\Sigma_\\alpha(-x,\\frac{1}{4})}{(1+x^2)^m}=\\frac{B(\\alpha,-x)}{(1+x^2)^m},\\quad \\frac{1}{12}<m\\leq \\frac{1}{6},\n\\]if $\\alpha =\\pm 1$.  \n\nThe asymptotics of the Pearcey integral can then be brought to bear to prove Proposition \\ref{dbu4th}.  In particular, one uses these asymptotics to show that the solution of the linear problem blows \nup at $(0,\\frac14)$, that it is continuous on the \ncomplement of  this point and that the relevant Duhamel integral is everywhere continuous. \nWe pass over the details.\n\\end{proof}\n\nThe analogous result for general dimensions $n\\in {\\mathbb N}$ can be established \nby passing to radial coordinates and reducing it to the one-dimensional case (as in \\cite{BAKS} where the first term in the asymptotic expansion is given).\n\n\n\n\n\n\n\\subsection{Third order NLS in one dimension}\n\nFinally, we briefly discuss the situation for a third-order nonlinear Schro\\\"odinger \nequation in $n=1$ dimensions. Consider  the initial=value problem\n\\begin{equation}\\label{HNLS}\n\\partial_t u +i\\alpha \\partial^2_{x} u +\\beta \\partial^3_{x} u +i\\gamma |u|^2u=0, \\quad u\\big|_{t=0}=u_0(x),\n\\end{equation}\nwhere $x\\in {\\mathbb R}$, and $\\alpha, \\beta, \\gamma \\in {\\mathbb R} \\setminus \\{ 0 \\}$ are  given parameters. \nThis is a  model problem that arises in optical wave propagation (see \\cite{Mu, T1}). \n\nEquation \\eqref{HNLS} appears similar to the well known Korteweg-de Vries equation, hence the \nappearance of dispersive blow up can be established along the same lines as was \npursued in \\cite{BS1} for this latter equation.  Indeed, \nthe associated linear equation,\n\\begin{equation}\\label{linHLS}\n\\partial_t  u +i\\alpha \\partial^2_{x}  u  +\\beta \\partial^3_{x}  u =0,\n\\end{equation}\nadmits a fundamental solution of the form\n\\[\n\\Lambda(x,t) = \\frac{1}{2\\pi} \\int_{\\mathbb R} e^{i\\beta \\xi^3 - i\\alpha \\xi^2 +ix \\cdot \\xi} \\, d\\xi.\n\\]\nThe expansion \n\\[\n\\left(\\xi+\\frac{\\alpha}{3\\beta}\\right )^3=\\xi^3+\\frac{\\alpha}{\\beta} \\xi^2+\\frac{\\alpha^2 \\xi}{3\\beta^2}+\\frac{\\alpha^3}{27 \\beta^3}\n\\]\nallows us to write $\\Lambda$ in the form\n\\[\n\\Lambda(x,t)=\\frac{1}{2\\pi (t\\beta)^{1\/3}}\\exp \\left(\\frac{4it\\alpha^3}{27 \\beta^2}\\right)\\exp\\left(\\frac{-i\\alpha x}{2\\beta^2}\\right)\\text{Ai}\\left(\\frac{1}{t^{1\/3} \\beta^{1\/3}}\n\\Big(x-\\frac{\\alpha ^2}{3\\beta}t\\Big)\\right).\n\\]\nHere, $\\text{Ai}$ denotes the well-known the Airy function defined, for example, as \nthe improper Riemann integral \n\\[\n\\text{Ai}(z)=\\int_{{\\mathbb R}}e^{i(\\xi^3+iz\\xi)}\\, d\\xi.\n\\]\nIn \\cite{BS1}, the dispersive properties of the Airy function are the basis for establishing dispersive blow up for the Korteweg-de Vries equation.  \nFollowing these ideas, dispersive blow up for \\eqref{linHLS} at $(x_*,t_*)=(0,1)$ is easily obtained by taking initial data of the form\n\\[\nu_0(x)=\\frac{A(-x)}{(1+x^2)^m}, \\quad \\text{with $\\frac{1}{8}<m\\leq \\frac{1}{4}$}\n\\]\nand\n\\[\nA(x)=\\text{Ai}\\left (\\frac{\\alpha^2+x}{3\\beta^{4\/3}}\\right ).\n\\]\nThe proof of dispersive blow up for  this third-order  Schr\\\"odinger equation \ncan then be accomplished just as in \\cite{BS1}. \nIndeed, the proof is  easier since, contrary to the Korteweg-de Vries equation, \nthe Duhamel representation of \\eqref{HNLS} does \nnot involve any spatial derivatives of the dependent variable.   Consequently \nit can be shown to be bounded by using only  Strichartz estimates for \nthe linearized Korteweg-de Vries equation (see \\cite{BAKS2, Bo}).\n\n\n\\begin{remark}  Combining  the results of the foregoing sections allows one to deduce\n  dispersive blow up for \nnonlinear Schr\\\"odinger-type equations with anisotropic dispersion, such as\n\\[\ni \\partial_t u+\\alpha \\Delta u +i\\beta \\partial^3_{x_1} u+\\gamma \\partial^4_{x_1}u+|u|^pu=0,\n\\]\nwhere $\\alpha, \\beta, \\gamma\\in {\\mathbb R} \\setminus \\{ 0 \\}$. The Cauchy problem for this equation \nhas been studied in \\cite{Bo} (see \nalso \\cite{FES}).\n\\end{remark}\n\n\\begin{merci}\nWe tender thanks to an anonymous referee for her\/his {\\it very} helpful remarks.\n\\end{merci}\n\n\n\n\\bibliographystyle{amsplain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n \n\nIn robot motor control, developing responsive control policies that adapt to unforeseen environments is crucial to task success. These changes and unexpected situations can be intrinsic or extrinsic, such as robot damage, motor failure, varying friction, and external force disturbances. For robot locomotion, traditional approaches of planning and control require expert knowledge and accurate dynamics models and constraints of both the robot and the environment \\cite{fankhauser2018robust, chatzinikolaidis2020}, which are all subject to unforeseeable changes that are difficult to know beforehand. Moreover, even using data-efficient learning techniques such as Bayesian optimization to tune decision variables and control parameters, it can only achieve adaptation on a trial-by-trial basis \\cite{rai2018bayesian} and also require extensive computation \\cite{yuan2019} which is not able to respond to changes on the fly.\n\nRecent advances in Reinforcement-Learning (RL) lead to algorithms achieving human-like or animal-level performance in a range of difficult control tasks. Model-free RL can perform global search of control parameters and obtain globally optimal gaits while combined with walking pattern generation \\cite{dallali2012global}. Also, an RL-based feedback policy can achieve human-like bipedal walking by imitating human motion capture data \\cite{yang2020learning}. With simulation training including actuator properties, a model-free RL scheme can train different locomotion policies separately and deploy on a real quadrupedal robot \\cite{hwangbo2019learning}. By using a multi-expert learning, an hierarchical RL architecture can learn to fuse multiple motor skills and generate multimodal locomotion coherently on a real quadruped \\cite{yang2020multi}. However, in general, model-free RL algorithms have limited sample efficiency, resulting in long training time to produce viable policies. For example, it took a model-free RL algorithm 83 hours to achieve human performance on the Atari game suite, compared to 15 minutes for a human~\\cite{hessel_rainbow_2017}. Similarly, AlphaStar~\\cite{alphastarblog} used 200 years of equivalent real-time to reach expert human performance playing Starcraft II. \n\nOn the other hand, model-based approaches can achieve comparatively high performance while being more sample-efficient by several orders of magnitude, converging faster than model-free approaches for locomotion tasks~\\cite{Chua}. To deploy robots in the real-world, online adaptation to changes in the environment are required as not all conditions can be considered by pre-trained policies, such as drastic changes of environments or robots by amputation. Hence, meta-learning, or learning to learn is a novel and more promising approach for solving such generic adaptations. Model-based meta-RL has been used in the real-world to adapt the control of a six-leg millirobot to different floor conditions~\\cite{GrBAL}. A model-based meta-RL algorithm, FAMLE, was used in the real-world on a Minitaur quadruped, where a latent black-box context vector encoded different environment conditions \\cite{FAMLE}. \n\n\n\\begin{figure}[t]\n    \\centering\n    \\begin{subfigure}{.2\\textwidth}\n      \\centering\n      \\includegraphics[width=\\textwidth]{external_force.png}\\hfill\n      \\caption{}\n    \\end{subfigure}\n    \\begin{subfigure}{.2\\textwidth}\n      \\centering\n      \\includegraphics[width=\\textwidth]{slippery_floor.png}\\hfill\n      \\caption{}\n    \\end{subfigure}\n     \\begin{subfigure}{.2\\textwidth}\n      \\centering\n      \\includegraphics[width=\\textwidth]{blocked_motor.png}\\hfill\n      \\caption{}\n    \\end{subfigure}\n    \\begin{subfigure}{.2\\textwidth}\n      \\centering\n      \\includegraphics[width=\\textwidth]{missing_leg.png}\\hfill\n      \\caption{}\n    \\end{subfigure}\n    \\vspace{-2mm}\n    \\caption{Adaptive and robust locomotion against uncertainties: (a) external force, (b) slippery ground, (c) faulty motors, and (d) leg amputation.}\n    \\label{fig:quadruped}\n    \\vspace{-5mm}\n\\end{figure}\n\n\\begin{figure*}[t]\n    \\centering\n    \\includegraphics[width=1\\linewidth]{Schematic_meta.png}\n    \\caption{Schematic view of the model-based meta-RL control framework for legged locomotion.}\n    \\label{fig:schematic}\n\\end{figure*}\n\n\n\nOur proposed method has made new improvements that require no prior knowledge of specific gaits. For example, FAMLE relies on sinusoidal gaits and therefore needs to optimize the amplitudes and phases of sinusoidal patterns by the model predictive control at a low frequency of 0.5Hz. In our work, we directly sample in the joint space at a much higher frequency at 50Hz, and we further improve the sampling process by specifying constraints on velocity, acceleration and jerk of the desired joint trajectories. Our study extensively validated the capability of adaptation in simulated test scenarios with large variations in floor friction, external forces or unexpected damage to joints.\n\nBased on the interaction model, our method allows changing the reward function online and therefore is able to modify the behavior of the robot. For example, the learned controller can track a variable forward velocity, even it has been trained on a fixed desired velocity. Likelihood estimation with condition latent vectors allows the meta-model to adapt to already seen conditions. Meta-training should allow \"on the fly\" optimization to better adapt to the current unknown condition.\n\nIn this paper, we present an improved model-based meta-RL approach to quadruped locomotion that is capable of online adaption to changing environments, as shown in Fig.~\\ref{fig:quadruped}. The main contributions of this work are:\n\\begin{enumerate}\n    \\item The proposed algorithm is capable of learning from scratch and requires no prior knowledge of the type of gaits, such as periodic phases of leg movements. \n    \\item Our methods introduces and applies hard constraints of velocity, acceleration and jerk on the sampled actions during the search process. \n    \\item The capability and robustness of online adaption to changes in both the robot and the environment, such as external force disturbances, varying frictions, faulty motors and leg amputation. \n\\end{enumerate}\n\n\nThe remainder of the paper is organized as follows. We outline the background in Section~\\ref{sec:Background} and related work in Section~\\ref{sec:RL}. In Section~\\ref{sec:method}, we elaborate the methodology and technical details on the model-based RL algorithm and the improvements by meta-learning. Section~\\ref{sec:exp} presents extensive simulation validations, results and analysis. Finally, we conclude and suggest future work in Section~\\ref{sec:Conclusion}.\n\n\n\\section{Background}\n\\label{sec:Background}\nThis section presents the preliminaries of RL, Model Predictive Control (MPC) and meta-learning. \n\\subsection{Reinforcement Learning}\n\\label{sec:RL}\n\nIn reinforcement learning, the agent learns to solve a task in an unknown environment $\\mathrm{E}$, defined by a Markov decision process $(\\mathrm{S},\\mathrm{s}_0, \\mathrm{A},\\mathrm{f}_\\mathrm{E},\\mathrm{r})$ where $\\mathrm{S}$ is the set of continuous states of the environment $\\mathrm{E}$, $\\mathrm{s}_0$ the initial state, $\\mathrm{A}$ the set of continuous actions the agent can perform in the environment, $\\mathrm{f}_\\mathrm{E}: \\mathrm{S}\\times \\mathrm{A}\\times\\mathrm{S} \\rightarrow \\mathbb{R}$ the probabilistic transition function and $\\mathrm{r}: \\mathrm{S}\\times \\mathrm{A} \\rightarrow \\mathbb{R}$ the reward function. \n\nThe goal of the agent is to learn a policy $\\Pi_{\\theta}: \\mathrm{S} \\rightarrow \\mathrm{A}$, parameterized by $\\theta$, which decides which action to perform given the current state $\\mathrm{s}_\\mathrm{t}$ to \\textit{maximize the long-term reward} $\\mathrm{R}(\\mathrm{t})=\\underset{\\mathrm{i}=\\mathrm{t}}{\\overset{\\mathrm{t}+\\mathrm{H}}{\\sum}}\\gamma^{\\mathrm{i}-\\mathrm{t}}\\mathrm{r}(\\mathrm{s}_\\mathrm{i},\\Pi(\\mathrm{s}_\\mathrm{i}))$, where $\\mathrm{H}$ is the horizon and $\\gamma$ is the discount factor. \n\nModel-free RL focuses in directly learning such a policy, whereas model-based RL focuses on learning a model of the transition function -- the transition of states given the current state and actions  -- which can be used to train the policy with fictive transitions or with model predictive control.\n\n\n\\subsection{Model Predictive Control}\n\\label{sec:MPC}\nGiven the current state $\\mathrm{s}_\\mathrm{t}$ and a horizon $\\mathrm{H}$, Model Predictive Control (MPC) uses a forward model of dynamics to select an action sequence $\\mathrm{a}_{\\mathrm{t}:\\mathrm{t}+\\mathrm{H}}$ which maximizes the predicted cumulative reward $\\mathrm{R}(\\mathrm{t})$. The agent performs the first action $\\mathrm{a}_\\mathrm{t}$ from the action sequence and collects the resulting state $\\mathrm{s}_{\\mathrm{t}+1}$. The MPC then repeats such optimization and allows the agent to alleviate the possible error in the model prediction. Compared to model-free RL, we can change the reward \\textit{online} to control the agent's behavior using model-based RL in an MPC fashion. \n\n\\subsection{Meta-Learning}\n\\label{sec:ML}\nWe use meta-learning to train an agent to solve several tasks, where the neural network learns to adapt to several varying conditions, such as different floor frictions, the presence of external disturbances or having a damaged motor. For the neural network model, the initial set of weights $\\theta^*$ must be found, such that only a small number of gradient descent steps with little collected data in a unknown environment can produce effective adaptations. \n\n\\section{Related Work}\n\\label{sec:RL}\n\\subsection{Model-Free Deep Reinforcement Learning}\nProximal Policy Optimization~\\cite{PPO} has been used to train a model-free controller to control Minitaur in simulation, producing trotting and gallop gaits \\cite{SimToReal} on the real robot. Soft Actor-Critic (SAC)~\\cite{SAC} has also been used to train on the real Minitaur robot within two hours~\\cite{LearningToWalk, ha_learning_2020}. This limitation of sampling-efficiency motivates us to focus on model-based RL. \n\n\\subsection{Model-Based Deep Reinforcement Learning}\nThere are three types of model-based RL: learning to predict the expected return from a starting state distribution, for example using Bayesian Optimization~\\cite{BayesianOptimisation}; learning to predict the outcome from a given starting state and given policy~\\cite{noveltySearch, MapElite, IMGEP}; and learning to model the transition function using a forward dynamical model. Here, we use the third type of model.\n\nForward models of dynamics are either deterministic or probabilistic, where deterministic models can be linear models~\\cite{levineLinearModel} or neural networks~\\cite{WorldModel}, and probabilistic models estimate uncertainty for modeling stochastic environments or estimating the long-term prediction uncertainty. Gaussian Processes~\\cite{Pilco, Multi-dex} or Bayesian neural networks~\\cite{Gal2015DropoutB} can be used to scale the abilities of Gaussian Processes models to higher dimensional environments. \n\nFor locomotion tasks, model-based RL with a forward model can have the same performance as model-free methods, while requiring at least an order of magnitude less samples~\\cite{Chua}. An ensemble of feed-forward neural networks is used to model the forward dynamics of the environment with uncertainty estimation. MPC uses this uncertainty estimation to formulate a more robust control which alleviates early overfitting model-based RL. The same method has been used with meta-learning to adapt the control of a 6-leg real millirobot to different floors \\cite{GrBAL}. \n\n\\subsection{MPC and Meta-Learning}\nSeveral optimization methods have been used for model-based RL, for example, Model Predictive Path Integral~\\cite{MPPI}, random shooting \\cite{LearningToWalk} or Cross-entropy method~\\cite{CEM, Chua}. We use random shooting for the simplicity, easy parallelism and proven performances on real robots~\\cite{GrBAL}.  \n\nThere are two main methods: a meta-learner model outputs the set of initial weights  $\\theta^*$ of the learner \\cite{learningToLearn}, or $\\theta^*$ is optimized using a meta-loss, it can be gradient-descent~\\cite{MAML} or evolutionary strategies~\\cite{ES-ML}. Gaussian processes have been used~\\cite{ml-gp} but only for low dimension environments. Meta-RL has been used with model-free RL~\\cite{noRML}, model-based RL~\\cite{GrBAL} or a mix of both~\\cite{M3PO}. For model-based RL, gradient based meta-learning was shown to be more data-efficient, resulting in a better and faster adaptation~\\cite{GrBAL}. Hence, we use gradient-based meta-learning. \n\nFor increasing generalization and adaptation to unseen condition of the environment, an adversarial loss has been used \\cite{adMRL}. Other methods employ context variables \\cite{context-variable}, bias transformation \\cite{bias-transformation}, or condition latent vector~\\cite{FAMLE}, to learn different input of sub-parts of the model for different condition, and then adapt this sub-parts to the current condition. \n\n\n\\section{Methodology}\n\\label{sec:method}\nThis section presents details of the model-based RL algorithm as discussed in Section~\\ref{sec:MBRL} and meta-learning algorithm Section~\\ref{sec:meta-learning}. We highlight our improvements which results in new robot capability of robust and versatile walking without a predefined, parameterized gait.\n\n\\subsection{Model-Based Reinforcement-Learning algorithm}\n\\label{sec:MBRL}\nThe model-based RL algorithm runs at 50Hz, sending desired actions to PD controllers running at 250Hz to generate torques for physics simulation. The algorithm is composed of two main parts: the forward dynamics model, and MPC. Fig.~\\ref{fig:schematic} illustrates the schematics of the control framework. \n\n\\subsubsection{The Forward Model of Dynamics}\nWe use a fully-connected feed-forward neural network, with two hidden layers of 256 units using a ReLU (Rectified Linear Unit) activation function. It takes the concatenation of the current state and action ($\\mathrm{s}_{\\mathrm{t}},\\mathrm{a}_\\mathrm{t}$) as input, and learns to predict the difference in the resulting state: $\\Delta \\mathrm{s}_\\mathrm{t}= \\mathrm{s}_\\mathrm{t+1}-\\mathrm{s}_{\\mathrm{t}}$, which is a standard means to get the prediction $\\mathrm{s}_\\mathrm{t+1} = \\mathrm{s}_{\\mathrm{t}} + \\Delta \\mathrm{s}_\\mathrm{t}$. \n\nThe model parameter $\\theta$ is the set of weights of the connections between the units. It is optimized using the gradient-based optimizer Adam~\\cite{adam} on a dataset of triplet $\\mathrm{D}\\in S\\times A \\times S$ using mean squared error as loss function. We depict details of the model-based RL algorithm in Algorithm~\\ref{algo:MBRL}.\n\nCompared to the work in~\\cite{FAMLE} for the Minitaur, our study formulate the state space as: the angular joints positions and velocities, the base orientation angles and angular rates, and the linear base velocities. The addition of the angular velocity of the base is the key of our success for controlling the robot at 50Hz. In contrast, only Euler angles and angular rates of the base in the horizontal plane were used in~\\cite{FAMLE} to control the Minitaur gait parameters at a much lower frequency of 0.5Hz. \n\n\\begin{algorithm}[t]\n\\caption{Model-based reinforcement learning algorithm \\newline Input: Environment $\\mathrm{E}$.}\n\\label{algo:MBRL}\n\\begin{algorithmic}\n\\STATE $\\mathrm{D}$ = $\\mathrm{N}_{init}$ episodes in $\\mathrm{E}$ with a random controller\n\\STATE $\\theta=$ random weights\n\\FOR{$\\mathrm{N}_\\text{training}$ episodes}\n    \\STATE Train $\\mathrm{M}_\\theta$ on $\\mathrm{D}$ using Adam\n    \\STATE $\\mathrm{s}_0=$ $\\mathrm{E}$.reset()\n    \\FOR{$\\mathrm{t}=0$ to $\\mathrm{T}-1$ steps}\n        \\STATE $\\mathrm{a}_\\mathrm{t}=$ MPC($\\mathrm{M}_\\theta$, $\\mathrm{s}_{t}$, $\\mathrm{a}_{\\mathrm{t}-3:\\mathrm{t}-1}$)\n        \\STATE $\\mathrm{s}_\\mathrm{t+1}=$ $\\mathrm{E}$.step($\\mathrm{a}_\\mathrm{t}$)\n        \\STATE $\\mathrm{D}=\\mathrm{D}\\cup \\{( \\mathrm{s}_{\\mathrm{t}}, \\mathrm{a}_\\mathrm{t}, \\mathrm{s}_\\mathrm{t+1})\\}$\n    \\ENDFOR\n\\ENDFOR\n\\end{algorithmic}\n\\end{algorithm}\n\n\\subsubsection{Model Predictive Control}\nThe method of random shooting is implemented which is suitable for parallel computing, and the algorithm is detailed in Algorithm~\\ref{algo:MPC}. At each time step, $\\mathrm{N}_\\text{pop}$ action sequences of length $\\mathrm{H}$ are sampled. Each sequence is evaluated starting with the current state, using the model to estimate the corresponding state trajectory. From these trajectories, long term reward is computed and the action with the highest estimated reward is selected.\n\nReal actuators have inherent limitations in velocity, acceleration and jerk. Instead of uniformly sampling desired joint angles within the limits, continuity constraints are used, where each desired joint state of the sequence is sampled using previous joint positions to ensure velocities, accelerations and jerks are smooth and bellow their respective limits. \n\nAs the improvement to the previous work~\\cite{FAMLE}, we enforced physical constraints during sampling of actions: $\\mathrm{q}_{min}\\le\\mathrm{q}\\le\\mathrm{q}_{max}$, $|\\mathrm{A}|\\le \\mathrm{A}_{max}$, $|\\mathrm{V}|\\le \\mathrm{V}_{max}$ and $|\\mathrm{J}|\\le \\mathrm{J}_{max}$, where $\\mathrm{q}$, $\\mathrm{V}$, $\\mathrm{A}$ and $\\mathrm{J}$ are the desired joint angle, velocity, acceleration and jerk, respectively. The limits of velocity, acceleration and jerk are the soft constraints for the smoothness and continuity of actions. For safety reasons, regarding the joint position limits, we further imposed hard constraints of sampled actions on $\\mathrm{q}$ to avoid hitting the physical limit of joint movements. \n\nThis improvements on sampling enforces the MPC in a more suitable subspace. During training, it increased the distance traveled compared to the default condition during a 10s episode by an order of 2: from $1.64\\pm0.12$m (without), to $3.16\\pm0.12$m (with), where p-value $\\le$ 0.001 on 20 episodes. It also reduces the observed jerk by an order of 5: from $1.3\\times {10}^{4} \\pm 260$ rad\/$\\mathrm{s}^3$ (without) to $2.4 \\times 10^3 \\pm 60$ rad\/$\\mathrm{s}^3$ (with), where p-value $\\le$ 0.001 on 20 episodes.\n\n\\begin{algorithm}[t]\n\\caption{MPC algorithm \\newline Inputs: A model $\\mathrm{M}_\\theta$, initial state $\\mathrm{s}_0$, past actions $\\mathrm{a}_{-3:-1}$, horizon $\\mathrm{H}$ and discount factor $\\gamma$.}\n\\label{algo:MPC}\n\\begin{algorithmic}\n\\STATE Sample $\\mathrm{a}_{1:\\mathrm{H}}^{1:\\mathrm{N}_\\text{pop}}$ using $\\mathrm{a}_{-3:-1}$ for continuity\n\\FOR{$\\mathrm{i}=1$ to $\\mathrm{N}$}\n    \\STATE $\\mathrm{R}^{\\mathrm{i}} = 0$\n\\ENDFOR\n\\FOR{$\\mathrm{t}=1$ to $\\mathrm{H}$ steps}\n    \\FORALL{$\\mathrm{i}=1$ to $\\mathrm{N}_\\text{pop}$ samples}\n        \\STATE $\\mathrm{s}^{\\mathrm{i}}_\\mathrm{t+1}=\\mathrm{M}_\\theta(a_{\\mathrm{t}}^{\\mathrm{i}}, \\mathrm{s}_{\\mathrm{t}})$\n        \\STATE $\\mathrm{R}^\\mathrm{i} = \\mathrm{R}^\\mathrm{i} + \\gamma ^{\\mathrm{t}-1}\\mathrm{r}(\\mathrm{s}^{\\mathrm{i}}_{\\mathrm{t}}, \\mathrm{a}^\\mathrm{i}_{\\mathrm{t}})$\n    \\ENDFOR\n\\ENDFOR\n\\STATE $\\mathrm{i}^*=\\underset{\\mathrm{i}}{\\mathrm{argmax}}$ $\\mathrm{R}^\\mathrm{i}$\n\\RETURN $\\mathrm{a}_1^{\\mathrm{i}^*}$\n\\end{algorithmic}\n\\end{algorithm}\n\n\\subsection{Meta-learning algorithm}\n\\label{sec:meta-learning}\nBefore meta-training, an expert is trained for each training condition using the proposed model-based RL algorithm to collect its training data. To adapt the model to each condition $\\mathrm{i}$, a specific latent vector $\\mathcal{C}_\\mathrm{i}$ is optimized during meta-learning using the regression loss on the data of the corresponding condition. This vector of fixed dimensions is then given to the input layer, alongside the current state and action when the condition $\\mathrm{i}$ is selected. We use a first order meta-learning called Reptile~\\cite{reptile}, which is composed of two phases: meta-training (Algorithm~\\ref{algo:meta_training}) and meta-adaptation (Algorithm~\\ref{algo:meta_adaptation}). \n\n\\subsubsection{Meta-training}\n\n\\begin{algorithm}[t]\n\\caption{Meta-learning training, called once before adaptation \\newline Inputs: $\\mathrm{N}$ datasets $\\mathrm{D}_{0:\\mathrm{N}-1}$ from $\\mathrm{N}$ different conditions.}\n\\label{algo:meta_training}\n\\begin{algorithmic}\n\\STATE $\\theta^*$ = random weights\n\\FOR{$\\mathrm{i}=0$ to $\\mathrm{N}-1$}\n    \\STATE $\\mathcal{C}_{\\mathrm{i}}$ = random vector\n\\ENDFOR\n\\FOR{$\\mathrm{n}_{outer}=0$ to $\\mathrm{N}_{outer}-1$}\n    \\STATE $\\mathrm{i} = \\mathrm{n}_{outer} \\mod \\mathrm{N}$\n    \\STATE $\\theta = \\theta^*$\n    \\FOR{$\\mathrm{n}_{inner}=1$ to $\\mathrm{N}_{inner}$}\n        \\STATE $\\mathcal{L} = \\tfrac{1}{|\\mathrm{D}_{\\mathrm{i}}|}\\underset{(\\mathrm{s},\\mathrm{a},\\mathrm{s}')\\in \\mathrm{D}_\\mathrm{i}}{\\sum} (\\mathrm{M}_\\theta(\\mathrm{s},\\mathrm{a}, \\mathcal{C}_{\\mathrm{i}})-\\mathrm{s}')^2$ \n        \\STATE $\\theta= \\theta - \\alpha \\nabla_\\theta \\mathcal{L}$ \n        \\STATE $\\mathcal{C}_{\\mathrm{i}}= \\mathcal{C}_{\\mathrm{i}} - \\alpha \\nabla_{\\mathcal{C}_{\\mathrm{i}}} \\mathcal{L}$ \n    \\ENDFOR\n    \\STATE $\\theta^* = \\theta^* + (1-\\tfrac{\\mathrm{n}_{outer}}{\\mathrm{N}_{outer}})\\beta (\\theta-\\theta^*)$ \n\\ENDFOR\n\\RETURN $\\theta^*$\n\\end{algorithmic}\n\\end{algorithm}\n\n\\begin{algorithm}[t]\n\\caption{Meta-learning adaptation, called at each step \\newline Inputs: a meta-trained set of weights $\\theta^*$, a list of learned $\\mathrm{N}$ condition latent vectors $\\mathcal{C}_{1:\\mathrm{N}}$, a dataset $\\mathrm{D}$ of the past $\\mathrm{K}$ steps.}\n\\label{algo:meta_adaptation}\n\\begin{algorithmic}\n\n\\FOR{$\\mathrm{i}=1$ to $\\mathrm{N}$}\n    \\STATE $\\mathcal{L}_\\mathrm{i} = \\tfrac{1}{|\\mathrm{D}|}\\underset{(\\mathrm{s},\\mathrm{a},\\mathrm{s}')\\in \\mathrm{D}}{\\sum} (\\mathrm{M}_\\theta^*(\\mathrm{s},\\mathrm{a}, \\mathcal{C}_\\mathrm{i})-\\mathrm{s}')^2$\n\\ENDFOR\n\\STATE $i^*= \\underset{\\mathrm{i}}{\\operatorname{argmin}} \\mathcal{L}_i$\n\\STATE $\\theta = \\theta^*$\n\\FOR{$n_{inner}=1$ to $\\mathrm{N}_{inner}$}\n    \\STATE $\\mathcal{L} = \\tfrac{1}{|\\mathrm{D}|} \\underset{(\\mathrm{s},\\mathrm{a},\\mathrm{s}')\\in D}{\\sum} (M_\\theta(\\mathrm{s},\\mathrm{a}, \\mathcal{C}_{\\mathrm{i^*}})-\\mathrm{s}')^2$\n    \\STATE $\\theta = \\theta - \\alpha \\nabla_\\theta \\mathcal{L}$ \n    \\STATE $\\mathcal{C}_{\\mathrm{i^*}} = \\mathcal{C}_{\\mathrm{i^*}} - \\alpha \\nabla_{\\mathcal{C}_{\\mathrm{i^*}}} \\mathcal{L}$ \n\\ENDFOR\n\\RETURN $\\theta$, $\\mathcal{C}_{\\mathrm{i^*}}$\n\\end{algorithmic}\n\\end{algorithm}\n\nThe initial set of weights $\\theta^*$ and each condition latent vector $\\mathcal{C}_\\mathrm{i}$ are optimized for adaptation. Meta-training is separated into two nested loops. In the inner loop, one training dataset $\\mathrm{D}_{\\mathrm{i}}$ and its corresponding condition latent vector $\\mathcal{C}_\\mathrm{i}$ are selected. The model weights $\\theta$ are initialized to $\\theta^*$ and Adam~\\cite{adam} optimize both of them for the regression loss of the current dataset $\\mathrm{D}_{\\mathrm{i}}$.\n\nIn the outer loop, $\\theta^*$ is optimized by taking a small step, with a linearly decreasing schedule, towards the optimized weights of the inner-loop. This allows $\\theta^*$ to converge to a nearby point (in the euclidean sense) to the optimal set of weights of each training condition. We detail the algorithm in Algorithm~\\ref{algo:meta_training}.\n\n\n\\subsubsection{Meta-adaptation}\nAt each time step, we select the most likely training condition using the previous $\\mathrm{K}$ time steps, each condition latent vector and the set of weights $\\theta^*$. \nWe then optimize the corresponding latent vector and the set of model weights, starting from $\\theta^*$, using the same optimization procedure as the inner loop but with the past $\\mathrm{K}$ steps. We detail the algorithm in Algorithm~\\ref{algo:meta_adaptation}.\n\nAfter the set of weights and the condition latent vector are optimized for the current condition, we use the MPC to select the optimal action to apply, and then new state information is collected, and the whole meta-adaptation iterates. This procedure allows any changes in the condition to be detected, and therefore the agent can adapt accordingly. \n\n\\subsection{Limitations}\nApart from the standard classical robot control of tuning PD gains and joints limits, the proposed method still requires fine-tuning of reward function, model architecture, hyper-parameters for the meta-learning and the adaptation. The use of MPC instead of a neural network policy has a trade-off between real-time computation and performance, i.e. MPC performs better in terms of adaption but requires more computation needed from the sampling procedure. \n\n\n\\begin{figure}[t]\n    \\centering\n      \\begin{subfigure}{\\linewidth}\n      \\centering\n      \\includegraphics[width=\\linewidth]{force_snapshots.png}\n      \\caption{}\n    \\end{subfigure}\n    \\begin{subfigure}{\\linewidth}\n      \\centering\n      \\includegraphics[width=\\linewidth]{slip_snapshots.png}\n      \\caption{}\n    \\end{subfigure}\n        \\begin{subfigure}{\\linewidth}\n      \\centering\n      \\includegraphics[width=\\linewidth]{damage_snapshots.png}\n      \\caption{}\n    \\end{subfigure}\n    \\begin{subfigure}{\\linewidth}\n      \\centering\n      \\includegraphics[width=\\linewidth]{tripod_snapshots.png}\n      \\caption{}\n    \\end{subfigure}\n    \\caption{Walking in presence of large uncertainties: (a) constant external force disturbance, (b) low-friction slippery ground, (c) faulty motors, and (d) with one missing leg.}\n    \n    \\vspace{-0mm}\n    \\label{fig:snapshots}\n\\end{figure}\n\n\n\\section{Results}\n\\label{sec:exp}\nWe used a custom version of the robot model (adapted from the open source SpotMicro robot~\\cite{kim_2019}) in PyBullet simulation to validate our method. Here, we first present the learning capability of the model-based RL algorithm on SpotMicro with a first adaption to a sequence of different conditions (Section~\\ref{sec:general_results}). We further validate the adaptation capability of the proposed meta-learning under various fixed frictions (Section~\\ref{sec:fixed_friction}) and time-varying, decreasing friction (Section~\\ref{sec:decreasing_friction}). \n\n\n\\subsection{Overview}\n\\label{sec:general_results}\n\n\\begin{figure}[t]\n    \\centering\n      \\begin{subfigure}{\\linewidth}\n      \\centering\n      \\includegraphics[width=\\linewidth]{velocity_snapshots_forward.png}\n      \\caption{}\n    \\end{subfigure}\n    \\begin{subfigure}{\\linewidth}\n      \\centering\n      \\includegraphics[width=\\linewidth]{velocity_snapshots_static.png}\n      \\caption{}\n    \\end{subfigure}\n        \\begin{subfigure}{\\linewidth}\n      \\centering\n      \\includegraphics[width=\\linewidth]{velocity_snapshots_backward.png}\n      \\caption{}\n    \\end{subfigure}\n    \\caption{Walking with a continuously changing velocity from 0.5m\/s to -0.2m\/s: (a) forward, (b) static, and (c) backward.}\n    \\vspace{-0mm}\n    \\label{fig:varying_velocity_snapshots}\n\\end{figure}\n\n\n\\begin{figure*}[t]\n    \\centering\n    \\includegraphics[width=\\linewidth]{only_left_side_joints.png} \n    \\caption{Measured joints trajectories of SpotMicro from a 20s test scenario where the robot or environment changed every 5s: default friction $\\mu=0.8$ (\\textcolor{YellowGreen}{\\textbf{green}}), blocked right hip pitch joint (\\textcolor{Lavender}{\\textbf{red}}), slippery ground (\\textcolor{Dandelion}{\\textbf{yellow}}) and constant external push (\\textcolor{Bittersweet}{\\textbf{brown}}).}\n    \\vspace{-2mm}\n    \\label{fig:joints_trajectories}\n\\end{figure*}\n\n\n\\begin{table*}[t]\n\\centering\n\\begin{tabular}{|l|c|c|c|c|}\n\\hline\nExpert\\textbackslash{}Condition & Default           & Slippery         & Lateral Force     & Damaged Motor      \\\\ \\hline\nDefault                         & 100\\%, 3.2$\\pm$ 0.2 & 20\\%, 0.7$\\pm$ 0.7 & 40\\%, 2.0$\\pm$ 1.0  & 100\\%, 0.7$\\pm$ 0.2  \\\\ \\hline\nSlippery                        & 30\\%, 1.3$\\pm$ 0.5  & 90\\%, 2.4$\\pm$ 0.4 & 10\\%, 0.7$\\pm$ 0.4  & 100\\%, 0.4$\\pm$  0.1 \\\\ \\hline\nLateral Force                   & 0\\%                & 0\\%                & 70\\%, 2.3 $\\pm$ 0.7 & 90\\%, 0.4$\\pm$ 0.1   \\\\ \\hline\nDamage Motor                    & 90\\%, 0.5$\\pm$ 0.2  & 70\\%, 0.3$\\pm$ 0.2 & 30\\%, 0.6$\\pm$ 0.3  & 100\\%, 2.5$\\pm$ 0.1  \\\\ \\hline\nMeta-Trained                    & 70\\%, 3.0$\\pm$ 0.6  & 80\\%, 2.4$\\pm$ 0.7 & 40\\%, 2.0$\\pm$ 1.3  & 100\\%, 2.7$\\pm$ 0.1  \\\\ \\hline\n\\end{tabular}\n\\caption{Success rate and average distance of travel for 10 episodes from different expert and the meta-trained model.}\n\\label{tab:success_rates}\n\\end{table*}\n\nWe trained the expert model with a default condition of friction $\\mu=0.8$, and this resulted default controller for walking is robust to perturbations, withstanding several pushes of 10N for 0.2s. After 300 of 10s-episodes which produced training data in the given condition, the quadruped was able to walk on slippery ground (friction coefficient $\\mu = 0.2$), against external forces or with a blocked motor or a missing\/amputated leg. Tab.~\\ref{tab:success_rates} shows the comparison between experts and meta-trained models under different conditions.\n\nUsing the proposed meta-learning method, the agent was able to adapt to four different conditions: default, fixed front-right hip motor, slippery ground and external forces. It traveled an average distance of $5.1\\pm1.5$m compared to a default expert which traveled $2.6\\pm0.8$m (averaged over 20 episodes). The joint trajectories from these test scenarios are shown in Fig.~\\ref{fig:joints_trajectories}.\n\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\linewidth]{varying_velocity.png}\n    \\caption{Online tracking of continuous and variable walking velocity using the model trained only at the fixed velocity of 0.5m\/s, while the desired velocity in the reward function changes from from 0.5m\/s to -0.2m\/s during the test scenario.}\n    \\vspace{-2mm}\n    \\label{fig:varying_velocity}\n\\end{figure}\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\linewidth]{random_velocity.png}\n    \\vspace{-4mm}\n    \\caption{Online tracking of discrete and variable walking velocity using the model trained only at the fixed velocity of 0.5m\/s, while every 2 s the desired velocity is randomly sampled within 0.5m\/s and -0.2m\/s during the test scenario.}\n    \\vspace{-0mm}\n    \\label{fig:random_velocity}\n\\end{figure}\n\n\nThe controller can achieve variable walking speed, despite being trained with only at a constant desired forward velocity, we can command different desired velocities online continuously, as shown in Fig.~\\ref{fig:varying_velocity_snapshots} and Fig.~\\ref{fig:varying_velocity}. Moreover, the trained expert controller can also generate continuous control actions to track discrete, discontinuous commanded velocities (see Fig. \\ref{fig:random_velocity}). \n\n\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\linewidth]{All_on_each_friction.png}\n    \\vspace{-0mm}\n    \\caption{Distance traveled for the full range of frictions from 3 expert models and the meta-trained model.}\n    \\vspace{-2mm}\n    \\label{fig:fixed_friction_adaptation}\n\\end{figure}\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\linewidth]{changing_friction_experts_VS_meta-learning_0.04s.png}\n    \\vspace{-0mm}\n    \\caption{Distance traveled with decreasing friction from the 3 expert models and the meta-trained model (p-value: $*<0.05$, $**<0.01$ and $***<0.001$). }\n    \\vspace{-2mm}\n    \\label{fig:changing_friction_adaptation}\n\\end{figure}\n\n\n\\subsection{Ground with Constant Friction}\n\\label{sec:fixed_friction}\nWe evaluated the adaptation capability using the meta-trained model and compared it to experts over the full range of different frictions (0.1 to 0.8 with 0.05 increments). We first trained 5 sets of experts for frictions 0.2, 0.4 and 0.6, using 300 10s-episodes, i.e., 50 minutes of data. Then we meta-trained 5 meta-models to adapt to these 3 frictions, each using one set of experts data, with the purpose that they could adapt to the full range afterwards.\n\nEach set of these 5 models was evaluated for each friction with 4 10s-episodes, so this gives 20 evaluations per expert and 20 evaluations for the meta-learning. The meta-trained models outperformed the experts on the full range of frictions, see Fig.~\\ref{fig:fixed_friction_adaptation}. As expected, each expert had its best performance when the friction constant is around its trained value.  \n\n\n\\subsection{Ground with Decreasing Friction}\n\\label{sec:decreasing_friction}j\n\nAs a comparison, we used the same experts and meta-model and benchmarked thei adaptation capability on a ground with continuously decreasing friction. We evaluated each set of models with 4 10s-episodes where friction coefficient $\\mu$ started at 0.8 and linearly decreased to 0.1. This gave 20 evaluations per expert and 20 evaluations for meta-learning.\n\nThe meta-trained models demonstrated better walking performance and traversed farther (3.38m) than the experts (3.07m), using a t-test with a p-value under $0.05$, see Fig.~\\ref{fig:changing_friction_adaptation}. In Fig.~\\ref{fig:velocity}, the curves and shaded areas are the means and standard deviations of the velocity, respectively. Snapshots of the walking gait using the meta-trained model are shown in Fig.~\\ref{fig:snapshots}, more details of walking performance can be seen in \\href{https:\/\/youtu.be\/s9OWhxorVmc}{the video here}. \n\n\nAdditionally, Fig.~\\ref{fig:condition_estimation_changing_friction} depicts the estimated condition at each time step using the past 0.1s (i.e. 5 time steps). At the beginning of the episode, when friction was higher, the model estimated a friction of 0.6 to be more likely (from 0.8 down to 0.5, i.e., 0-4.5s), then switched to a friction of 0.4 (from 0.5 down to 0.3, i.e., 4.5-7s), and finished by estimating a more likely friction of 0.2 (from 0.3 down to 0.1, i.e., 7-10s).\n\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\linewidth]{velocity_decreasing_frictions_all.png}\n    \\vspace{-4mm}\n    \\caption{Forward walking velocity with decreasing friction from 3 expert models and the meta-trained model.}\n    \\vspace{-0mm}\n    \\label{fig:velocity}\n\\end{figure}\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\linewidth]{likelyhood_estimation_0.1s.png}\n    \\vspace{-4mm}\n    \\caption{Likelihood of each pre-trained condition using online meta-adaptation in presence of decreasing friction.}\n    \\vspace{-0mm}\n    \\label{fig:condition_estimation_changing_friction}\n\\end{figure}\n\n\\subsection{Discussion}\nThe results validate the efficiency and effectiveness of the meta-learning method to detect the most probable current condition and adapt accordingly to different ground friction coefficients. The online adaptive walking using the proposed meta-learning outperformed the specialized experts which were specifically trained on specific frictions. This demonstrated the capability of meta-learning to incorporate knowledge from all training data.\n\nWe have also pushed the extreme test case in terms of unseen hardware failures. We specifically designed a case for meta-training where a motor of one leg of the quadruped was blocked at a fixed joint position (emulated actuator failures), and tested if meta-learning can adapt to the damage from a different leg faster than learning from scratch. Our investigation showed that the meta-adaptation is not able to adapt to such changes on a different leg, and we hypothesize that a second-order meta-learning algorithm, e.g. Model-Agnostic Meta-Learning~\\cite{MAML}, may be a better solution for such an extreme case.\n\n\\section{Conclusion and future work}\n\\label{sec:Conclusion}\nBased on the past work of model-based meta-RL~\\cite{FAMLE}, we have made contributions to improve the algorithm for adaptive and robust quadruped locomotion in changing robot dynamics (motor failure and amputation) and varying environmental constraints (time-varying friction, external pushes). In physics-based simulation, we have demonstrated this method can learn quadrupedal walking without using a periodic gait signal \\cite{FAMLE} or a phase vector \\cite{yang2020learning}. Instead, by updating an interaction model of the robot and environment and applying the optimal control actions, a walking gait is naturally generated as the outcome of maximizing the task reward. We further validated the capability of our proposed framework in adapting to different conditions such as robot damage, changing friction, and external force disturbances. \n\nFuture work will apply this method on the real SpotMicro robot, and identify potential issues of sim2real transfer which will be addressed by new solutions for meta-model and more effective search of model predictive control procedure. We hypothesize the current meta-learning algorithm is efficient enough for multi-task learning, which can be further studied. Also, a second-order meta-learning algorithm~\\cite{MAML} can be Incorporated could potentially achieve better adaptation to novel situations. \n\n\n\\section{Acknowledgement}\nThis work has been supported by EPSRC UK Robotics and Artificial Intelligence Hub for Offshore Energy Asset Integrity Management (EP\/R026173\/1).\n\n\n\\bibliographystyle{IEEEtran}\n\\balance\n\n\\section{Introduction}\n \n\nIn robot motor control, developing responsive control policies that adapt to unforeseen environments is crucial to task success. These changes and unexpected situations can be intrinsic or extrinsic, such as robot damage, motor failure, varying friction, and external force disturbances. For robot locomotion, traditional approaches of planning and control require expert knowledge and accurate dynamics models and constraints of both the robot and the environment \\cite{fankhauser2018robust, chatzinikolaidis2020}, which are all subject to unforeseeable changes that are difficult to know beforehand. Moreover, even using data-efficient learning techniques such as Bayesian optimization to tune decision variables and control parameters, it can only achieve adaptation on a trial-by-trial basis \\cite{rai2018bayesian} and also require extensive computation \\cite{yuan2019} which is not able to respond to changes on the fly.\n\nRecent advances in Reinforcement-Learning (RL) lead to algorithms achieving human-like or animal-level performance in a range of difficult control tasks. Model-free RL can perform global search of control parameters and obtain globally optimal gaits while combined with walking pattern generation \\cite{dallali2012global}. Also, an RL-based feedback policy can achieve human-like bipedal walking by imitating human motion capture data \\cite{yang2020learning}. With simulation training including actuator properties, a model-free RL scheme can train different locomotion policies separately and deploy on a real quadrupedal robot \\cite{hwangbo2019learning}. By using a multi-expert learning, an hierarchical RL architecture can learn to fuse multiple motor skills and generate multimodal locomotion coherently on a real quadruped \\cite{yang2020multi}. However, in general, model-free RL algorithms have limited sample efficiency, resulting in long training time to produce viable policies. For example, it took a model-free RL algorithm 83 hours to achieve human performance on the Atari game suite, compared to 15 minutes for a human~\\cite{hessel_rainbow_2017}. Similarly, AlphaStar~\\cite{alphastarblog} used 200 years of equivalent real-time to reach expert human performance playing Starcraft II. \n\nOn the other hand, model-based approaches can achieve comparatively high performance while being more sample-efficient by several orders of magnitude, converging faster than model-free approaches for locomotion tasks~\\cite{Chua}. To deploy robots in the real-world, online adaptation to changes in the environment are required as not all conditions can be considered by pre-trained policies, such as drastic changes of environments or robots by amputation. Hence, meta-learning, or learning to learn is a novel and more promising approach for solving such generic adaptations. Model-based meta-RL has been used in the real-world to adapt the control of a six-leg millirobot to different floor conditions~\\cite{GrBAL}. A model-based meta-RL algorithm, FAMLE, was used in the real-world on a Minitaur quadruped, where a latent black-box context vector encoded different environment conditions \\cite{FAMLE}. \n\n\n\\begin{figure}[t]\n    \\centering\n    \\begin{subfigure}{.2\\textwidth}\n      \\centering\n      \\includegraphics[width=\\textwidth]{external_force.png}\\hfill\n      \\caption{}\n    \\end{subfigure}\n    \\begin{subfigure}{.2\\textwidth}\n      \\centering\n      \\includegraphics[width=\\textwidth]{slippery_floor.png}\\hfill\n      \\caption{}\n    \\end{subfigure}\n     \\begin{subfigure}{.2\\textwidth}\n      \\centering\n      \\includegraphics[width=\\textwidth]{blocked_motor.png}\\hfill\n      \\caption{}\n    \\end{subfigure}\n    \\begin{subfigure}{.2\\textwidth}\n      \\centering\n      \\includegraphics[width=\\textwidth]{missing_leg.png}\\hfill\n      \\caption{}\n    \\end{subfigure}\n    \\vspace{-2mm}\n    \\caption{Adaptive and robust locomotion against uncertainties: (a) external force, (b) slippery ground, (c) faulty motors, and (d) leg amputation.}\n    \\label{fig:quadruped}\n    \\vspace{-5mm}\n\\end{figure}\n\n\\begin{figure*}[t]\n    \\centering\n    \\includegraphics[width=1\\linewidth]{Schematic_meta.png}\n    \\caption{Schematic view of the model-based meta-RL control framework for legged locomotion.}\n    \\label{fig:schematic}\n\\end{figure*}\n\n\n\nOur proposed method has made new improvements that require no prior knowledge of specific gaits. For example, FAMLE relies on sinusoidal gaits and therefore needs to optimize the amplitudes and phases of sinusoidal patterns by the model predictive control at a low frequency of 0.5Hz. In our work, we directly sample in the joint space at a much higher frequency at 50Hz, and we further improve the sampling process by specifying constraints on velocity, acceleration and jerk of the desired joint trajectories. Our study extensively validated the capability of adaptation in simulated test scenarios with large variations in floor friction, external forces or unexpected damage to joints.\n\nBased on the interaction model, our method allows changing the reward function online and therefore is able to modify the behavior of the robot. For example, the learned controller can track a variable forward velocity, even it has been trained on a fixed desired velocity. Likelihood estimation with condition latent vectors allows the meta-model to adapt to already seen conditions. Meta-training should allow \"on the fly\" optimization to better adapt to the current unknown condition.\n\nIn this paper, we present an improved model-based meta-RL approach to quadruped locomotion that is capable of online adaption to changing environments, as shown in Fig.~\\ref{fig:quadruped}. The main contributions of this work are:\n\\begin{enumerate}\n    \\item The proposed algorithm is capable of learning from scratch and requires no prior knowledge of the type of gaits, such as periodic phases of leg movements. \n    \\item Our methods introduces and applies hard constraints of velocity, acceleration and jerk on the sampled actions during the search process. \n    \\item The capability and robustness of online adaption to changes in both the robot and the environment, such as external force disturbances, varying frictions, faulty motors and leg amputation. \n\\end{enumerate}\n\n\nThe remainder of the paper is organized as follows. We outline the background in Section~\\ref{sec:Background} and related work in Section~\\ref{sec:RW}. In Section~\\ref{sec:method}, we elaborate the methodology and technical details on the model-based RL algorithm and the improvements by meta-learning. Section~\\ref{sec:exp} presents extensive simulation validations, results and analysis. Finally, we conclude and suggest future work in Section~\\ref{sec:Conclusion}.\n\n\n\\section{Background}\n\\label{sec:Background}\nThis section presents the preliminaries of RL, Model Predictive Control (MPC) and meta-learning. \n\\subsection{Reinforcement Learning}\n\\label{sec:RL}\n\nIn reinforcement learning, the agent learns to solve a task in an unknown environment $\\mathrm{E}$, defined by a Markov decision process $(\\mathrm{S},\\mathrm{s}_0, \\mathrm{A},\\mathrm{f}_\\mathrm{E},\\mathrm{r})$ where $\\mathrm{S}$ is the set of continuous states of the environment $\\mathrm{E}$, $\\mathrm{s}_0$ the initial state, $\\mathrm{A}$ the set of continuous actions the agent can perform in the environment, $\\mathrm{f}_\\mathrm{E}: \\mathrm{S}\\times \\mathrm{A}\\times\\mathrm{S} \\rightarrow \\mathbb{R}$ the probabilistic transition function and $\\mathrm{r}: \\mathrm{S}\\times \\mathrm{A} \\rightarrow \\mathbb{R}$ the reward function. \n\nThe goal of the agent is to learn a policy $\\Pi_{\\theta}: \\mathrm{S} \\rightarrow \\mathrm{A}$, parameterized by $\\theta$, which decides which action to perform given the current state $\\mathrm{s}_\\mathrm{t}$ to \\textit{maximize the long-term reward} $\\mathrm{R}(\\mathrm{t})=\\underset{\\mathrm{i}=\\mathrm{t}}{\\overset{\\mathrm{t}+\\mathrm{H}}{\\sum}}\\gamma^{\\mathrm{i}-\\mathrm{t}}\\mathrm{r}(\\mathrm{s}_\\mathrm{i},\\Pi(\\mathrm{s}_\\mathrm{i}))$, where $\\mathrm{H}$ is the horizon and $\\gamma$ is the discount factor. \n\nModel-free RL focuses in directly learning such a policy, whereas model-based RL focuses on learning a model of the transition function -- the transition of states given the current state and actions  -- which can be used to train the policy with fictive transitions or with model predictive control.\n\n\n\\subsection{Model Predictive Control}\n\\label{sec:MPC}\nGiven the current state $\\mathrm{s}_\\mathrm{t}$ and a horizon $\\mathrm{H}$, Model Predictive Control (MPC) uses a forward model of dynamics to select an action sequence $\\mathrm{a}_{\\mathrm{t}:\\mathrm{t}+\\mathrm{H}}$ which maximizes the predicted cumulative reward $\\mathrm{R}(\\mathrm{t})$. The agent performs the first action $\\mathrm{a}_\\mathrm{t}$ from the action sequence and collects the resulting state $\\mathrm{s}_{\\mathrm{t}+1}$. The MPC then repeats such optimization and allows the agent to alleviate the possible error in the model prediction. Compared to model-free RL, we can change the reward \\textit{online} to control the agent's behavior using model-based RL in an MPC fashion. \n\n\\subsection{Meta-Learning}\n\\label{sec:ML}\nWe use meta-learning to train an agent to solve several tasks, where the neural network learns to adapt to several varying conditions, such as different floor frictions, the presence of external disturbances or having a damaged motor. For the neural network model, the initial set of weights $\\theta^*$ must be found, such that only a small number of gradient descent steps with little collected data in a unknown environment can produce effective adaptations. \n\n\\section{Related Work}\n\\label{sec:RW}\n\\subsection{Model-Free Deep Reinforcement Learning}\nProximal Policy Optimization~\\cite{PPO} has been used to train a model-free controller to control Minitaur in simulation, producing trotting and gallop gaits \\cite{SimToReal} on the real robot. Soft Actor-Critic (SAC)~\\cite{SAC} has also been used to train on the real Minitaur robot within two hours~\\cite{LearningToWalk, ha_learning_2020}. This limitation of sampling-efficiency motivates us to focus on model-based RL. \n\n\\subsection{Model-Based Deep Reinforcement Learning}\nThere are three types of model-based RL: learning to predict the expected return from a starting state distribution, for example using Bayesian Optimization~\\cite{BayesianOptimisation}; learning to predict the outcome from a given starting state and given policy~\\cite{noveltySearch, MapElite, IMGEP}; and learning to model the transition function using a forward dynamical model. Here, we use the third type of model.\n\nForward models of dynamics are either deterministic or probabilistic, where deterministic models can be linear models~\\cite{levineLinearModel} or neural networks~\\cite{WorldModel}, and probabilistic models estimate uncertainty for modeling stochastic environments or estimating the long-term prediction uncertainty. Gaussian Processes~\\cite{Pilco, Multi-dex} or Bayesian neural networks~\\cite{Gal2015DropoutB} can be used to scale the abilities of Gaussian Processes models to higher dimensional environments. \n\nFor locomotion tasks, model-based RL with a forward model can have the same performance as model-free methods, while requiring at least an order of magnitude less samples~\\cite{Chua}. An ensemble of feed-forward neural networks is used to model the forward dynamics of the environment with uncertainty estimation. MPC uses this uncertainty estimation to formulate a more robust control which alleviates early overfitting model-based RL. The same method has been used with meta-learning to adapt the control of a 6-leg real millirobot to different floors \\cite{GrBAL}. \n\n\\subsection{MPC and Meta-Learning}\nSeveral optimization methods have been used for model-based RL, for example, Model Predictive Path Integral~\\cite{MPPI}, random shooting \\cite{LearningToWalk} or Cross-entropy method~\\cite{CEM, Chua}. We use random shooting for the simplicity, easy parallelism and proven performances on real robots~\\cite{GrBAL}.  \n\nThere are two main methods: a meta-learner model outputs the set of initial weights  $\\theta^*$ of the learner \\cite{learningToLearn}, or $\\theta^*$ is optimized using a meta-loss, it can be gradient-descent~\\cite{MAML} or evolutionary strategies~\\cite{ES-ML}. Gaussian processes have been used~\\cite{ml-gp} but only for low dimension environments. Meta-RL has been used with model-free RL~\\cite{noRML}, model-based RL~\\cite{GrBAL} or a mix of both~\\cite{M3PO}. For model-based RL, gradient based meta-learning was shown to be more data-efficient, resulting in a better and faster adaptation~\\cite{GrBAL}. Hence, we use gradient-based meta-learning. \n\nFor increasing generalization and adaptation to unseen condition of the environment, an adversarial loss has been used \\cite{adMRL}. Other methods employ context variables \\cite{context-variable}, bias transformation \\cite{bias-transformation}, or condition latent vector~\\cite{FAMLE}, to learn different input of sub-parts of the model for different condition, and then adapt this sub-parts to the current condition. \n\n\n\\section{Methodology}\n\\label{sec:method}\nThis section presents details of the model-based RL algorithm as discussed in Section~\\ref{sec:MBRL} and meta-learning algorithm Section~\\ref{sec:meta-learning}. We highlight our improvements which results in new robot capability of robust and versatile walking without a predefined, parameterized gait.\n\n\\subsection{Model-Based Reinforcement-Learning algorithm}\n\\label{sec:MBRL}\nThe model-based RL algorithm runs at 50Hz, sending desired actions to PD controllers running at 250Hz to generate torques for physics simulation. The algorithm is composed of two main parts: the forward dynamics model, and MPC. Fig.~\\ref{fig:schematic} illustrates the schematics of the control framework. \n\n\\subsubsection{The Forward Model of Dynamics}\nWe use a fully-connected feed-forward neural network, with two hidden layers of 256 units using a ReLU (Rectified Linear Unit) activation function. It takes the concatenation of the current state and action ($\\mathrm{s}_{\\mathrm{t}},\\mathrm{a}_\\mathrm{t}$) as input, and learns to predict the difference in the resulting state: $\\Delta \\mathrm{s}_\\mathrm{t}= \\mathrm{s}_\\mathrm{t+1}-\\mathrm{s}_{\\mathrm{t}}$, which is a standard means to get the prediction $\\mathrm{s}_\\mathrm{t+1} = \\mathrm{s}_{\\mathrm{t}} + \\Delta \\mathrm{s}_\\mathrm{t}$. \n\nThe model parameter $\\theta$ is the set of weights of the connections between the units. It is optimized using the gradient-based optimizer Adam~\\cite{adam} on a dataset of triplet $\\mathrm{D}\\in S\\times A \\times S$ using mean squared error as loss function. We depict details of the model-based RL algorithm in Algorithm~\\ref{algo:MBRL}.\n\nCompared to the work in~\\cite{FAMLE} for the Minitaur, our study formulate the state space as: the angular joints positions and velocities, the base orientation angles and angular rates, and the linear base velocities. The addition of the angular velocity of the base is the key of our success for controlling the robot at 50Hz. In contrast, only Euler angles and angular rates of the base in the horizontal plane were used in~\\cite{FAMLE} to control the Minitaur gait parameters at a much lower frequency of 0.5Hz. \n\n\\begin{algorithm}[t]\n\\caption{Model-based reinforcement learning algorithm \\newline Input: Environment $\\mathrm{E}$.}\n\\label{algo:MBRL}\n\\begin{algorithmic}\n\\STATE $\\mathrm{D}$ = $\\mathrm{N}_{init}$ episodes in $\\mathrm{E}$ with a random controller\n\\STATE $\\theta=$ random weights\n\\FOR{$\\mathrm{N}_\\text{training}$ episodes}\n    \\STATE Train $\\mathrm{M}_\\theta$ on $\\mathrm{D}$ using Adam\n    \\STATE $\\mathrm{s}_0=$ $\\mathrm{E}$.reset()\n    \\FOR{$\\mathrm{t}=0$ to $\\mathrm{T}-1$ steps}\n        \\STATE $\\mathrm{a}_\\mathrm{t}=$ MPC($\\mathrm{M}_\\theta$, $\\mathrm{s}_{t}$, $\\mathrm{a}_{\\mathrm{t}-3:\\mathrm{t}-1}$)\n        \\STATE $\\mathrm{s}_\\mathrm{t+1}=$ $\\mathrm{E}$.step($\\mathrm{a}_\\mathrm{t}$)\n        \\STATE $\\mathrm{D}=\\mathrm{D}\\cup \\{( \\mathrm{s}_{\\mathrm{t}}, \\mathrm{a}_\\mathrm{t}, \\mathrm{s}_\\mathrm{t+1})\\}$\n    \\ENDFOR\n\\ENDFOR\n\\end{algorithmic}\n\\end{algorithm}\n\n\\subsubsection{Model Predictive Control}\nThe method of random shooting is implemented which is suitable for parallel computing, and the algorithm is detailed in Algorithm~\\ref{algo:MPC}. At each time step, $\\mathrm{N}_\\text{pop}$ action sequences of length $\\mathrm{H}$ are sampled. Each sequence is evaluated starting with the current state, using the model to estimate the corresponding state trajectory. From these trajectories, long term reward is computed and the action with the highest estimated reward is selected.\n\nReal actuators have inherent limitations in velocity, acceleration and jerk. Instead of uniformly sampling desired joint angles within the limits, continuity constraints are used, where each desired joint state of the sequence is sampled using previous joint positions to ensure velocities, accelerations and jerks are smooth and bellow their respective limits. \n\nAs the improvement to the previous work~\\cite{FAMLE}, we enforced physical constraints during sampling of actions: $\\mathrm{q}_{min}\\le\\mathrm{q}\\le\\mathrm{q}_{max}$, $|\\mathrm{A}|\\le \\mathrm{A}_{max}$, $|\\mathrm{V}|\\le \\mathrm{V}_{max}$ and $|\\mathrm{J}|\\le \\mathrm{J}_{max}$, where $\\mathrm{q}$, $\\mathrm{V}$, $\\mathrm{A}$ and $\\mathrm{J}$ are the desired joint angle, velocity, acceleration and jerk, respectively. The limits of velocity, acceleration and jerk are the soft constraints for the smoothness and continuity of actions. For safety reasons, regarding the joint position limits, we further imposed hard constraints of sampled actions on $\\mathrm{q}$ to avoid hitting the physical limit of joint movements. \n\nThis improvements on sampling enforces the MPC in a more suitable subspace. During training, it increased the distance traveled compared to the default condition during a 10s episode by an order of 2: from $1.64\\pm0.12$m (without), to $3.16\\pm0.12$m (with), where p-value $\\le$ 0.001 on 20 episodes. It also reduces the observed jerk by an order of 5: from $1.3\\times {10}^{4} \\pm 260$ rad\/$\\mathrm{s}^3$ (without) to $2.4 \\times 10^3 \\pm 60$ rad\/$\\mathrm{s}^3$ (with), where p-value $\\le$ 0.001 on 20 episodes.\n\n\\begin{algorithm}[t]\n\\caption{MPC algorithm \\newline Inputs: A model $\\mathrm{M}_\\theta$, initial state $\\mathrm{s}_0$, past actions $\\mathrm{a}_{-3:-1}$, horizon $\\mathrm{H}$ and discount factor $\\gamma$.}\n\\label{algo:MPC}\n\\begin{algorithmic}\n\\STATE Sample $\\mathrm{a}_{1:\\mathrm{H}}^{1:\\mathrm{N}_\\text{pop}}$ using $\\mathrm{a}_{-3:-1}$ for continuity\n\\FOR{$\\mathrm{i}=1$ to $\\mathrm{N}$}\n    \\STATE $\\mathrm{R}^{\\mathrm{i}} = 0$\n\\ENDFOR\n\\FOR{$\\mathrm{t}=1$ to $\\mathrm{H}$ steps}\n    \\FORALL{$\\mathrm{i}=1$ to $\\mathrm{N}_\\text{pop}$ samples}\n        \\STATE $\\mathrm{s}^{\\mathrm{i}}_\\mathrm{t+1}=\\mathrm{M}_\\theta(a_{\\mathrm{t}}^{\\mathrm{i}}, \\mathrm{s}_{\\mathrm{t}})$\n        \\STATE $\\mathrm{R}^\\mathrm{i} = \\mathrm{R}^\\mathrm{i} + \\gamma ^{\\mathrm{t}-1}\\mathrm{r}(\\mathrm{s}^{\\mathrm{i}}_{\\mathrm{t}}, \\mathrm{a}^\\mathrm{i}_{\\mathrm{t}})$\n    \\ENDFOR\n\\ENDFOR\n\\STATE $\\mathrm{i}^*=\\underset{\\mathrm{i}}{\\mathrm{argmax}}$ $\\mathrm{R}^\\mathrm{i}$\n\\RETURN $\\mathrm{a}_1^{\\mathrm{i}^*}$\n\\end{algorithmic}\n\\end{algorithm}\n\n\\subsection{Meta-learning algorithm}\n\\label{sec:meta-learning}\nBefore meta-training, an expert is trained for each training condition using the proposed model-based RL algorithm to collect its training data. To adapt the model to each condition $\\mathrm{i}$, a specific latent vector $\\mathcal{C}_\\mathrm{i}$ is optimized during meta-learning using the regression loss on the data of the corresponding condition. This vector of fixed dimensions is then given to the input layer, alongside the current state and action when the condition $\\mathrm{i}$ is selected. We use a first order meta-learning called Reptile~\\cite{reptile}, which is composed of two phases: meta-training (Algorithm~\\ref{algo:meta_training}) and meta-adaptation (Algorithm~\\ref{algo:meta_adaptation}). \n\n\\subsubsection{Meta-training}\n\n\\begin{algorithm}[t]\n\\caption{Meta-learning training, called once before adaptation \\newline Inputs: $\\mathrm{N}$ datasets $\\mathrm{D}_{0:\\mathrm{N}-1}$ from $\\mathrm{N}$ different conditions.}\n\\label{algo:meta_training}\n\\begin{algorithmic}\n\\STATE $\\theta^*$ = random weights\n\\FOR{$\\mathrm{i}=0$ to $\\mathrm{N}-1$}\n    \\STATE $\\mathcal{C}_{\\mathrm{i}}$ = random vector\n\\ENDFOR\n\\FOR{$\\mathrm{n}_{outer}=0$ to $\\mathrm{N}_{outer}-1$}\n    \\STATE $\\mathrm{i} = \\mathrm{n}_{outer} \\mod \\mathrm{N}$\n    \\STATE $\\theta = \\theta^*$\n    \\FOR{$\\mathrm{n}_{inner}=1$ to $\\mathrm{N}_{inner}$}\n        \\STATE $\\mathcal{L} = \\tfrac{1}{|\\mathrm{D}_{\\mathrm{i}}|}\\underset{(\\mathrm{s},\\mathrm{a},\\mathrm{s}')\\in \\mathrm{D}_\\mathrm{i}}{\\sum} (\\mathrm{M}_\\theta(\\mathrm{s},\\mathrm{a}, \\mathcal{C}_{\\mathrm{i}})-\\mathrm{s}')^2$ \n        \\STATE $\\theta= \\theta - \\alpha \\nabla_\\theta \\mathcal{L}$ \n        \\STATE $\\mathcal{C}_{\\mathrm{i}}= \\mathcal{C}_{\\mathrm{i}} - \\alpha \\nabla_{\\mathcal{C}_{\\mathrm{i}}} \\mathcal{L}$ \n    \\ENDFOR\n    \\STATE $\\theta^* = \\theta^* + (1-\\tfrac{\\mathrm{n}_{outer}}{\\mathrm{N}_{outer}})\\beta (\\theta-\\theta^*)$ \n\\ENDFOR\n\\RETURN $\\theta^*$\n\\end{algorithmic}\n\\end{algorithm}\n\n\\begin{algorithm}[t]\n\\caption{Meta-learning adaptation, called at each step \\newline Inputs: a meta-trained set of weights $\\theta^*$, a list of learned $\\mathrm{N}$ condition latent vectors $\\mathcal{C}_{1:\\mathrm{N}}$, a dataset $\\mathrm{D}$ of the past $\\mathrm{K}$ steps.}\n\\label{algo:meta_adaptation}\n\\begin{algorithmic}\n\n\\FOR{$\\mathrm{i}=1$ to $\\mathrm{N}$}\n    \\STATE $\\mathcal{L}_\\mathrm{i} = \\tfrac{1}{|\\mathrm{D}|}\\underset{(\\mathrm{s},\\mathrm{a},\\mathrm{s}')\\in \\mathrm{D}}{\\sum} (\\mathrm{M}_\\theta^*(\\mathrm{s},\\mathrm{a}, \\mathcal{C}_\\mathrm{i})-\\mathrm{s}')^2$\n\\ENDFOR\n\\STATE $i^*= \\underset{\\mathrm{i}}{\\operatorname{argmin}} \\mathcal{L}_i$\n\\STATE $\\theta = \\theta^*$\n\\FOR{$n_{inner}=1$ to $\\mathrm{N}_{inner}$}\n    \\STATE $\\mathcal{L} = \\tfrac{1}{|\\mathrm{D}|} \\underset{(\\mathrm{s},\\mathrm{a},\\mathrm{s}')\\in D}{\\sum} (M_\\theta(\\mathrm{s},\\mathrm{a}, \\mathcal{C}_{\\mathrm{i^*}})-\\mathrm{s}')^2$\n    \\STATE $\\theta = \\theta - \\alpha \\nabla_\\theta \\mathcal{L}$ \n    \\STATE $\\mathcal{C}_{\\mathrm{i^*}} = \\mathcal{C}_{\\mathrm{i^*}} - \\alpha \\nabla_{\\mathcal{C}_{\\mathrm{i^*}}} \\mathcal{L}$ \n\\ENDFOR\n\\RETURN $\\theta$, $\\mathcal{C}_{\\mathrm{i^*}}$\n\\end{algorithmic}\n\\end{algorithm}\n\nThe initial set of weights $\\theta^*$ and each condition latent vector $\\mathcal{C}_\\mathrm{i}$ are optimized for adaptation. Meta-training is separated into two nested loops. In the inner loop, one training dataset $\\mathrm{D}_{\\mathrm{i}}$ and its corresponding condition latent vector $\\mathcal{C}_\\mathrm{i}$ are selected. The model weights $\\theta$ are initialized to $\\theta^*$ and Adam~\\cite{adam} optimize both of them for the regression loss of the current dataset $\\mathrm{D}_{\\mathrm{i}}$.\n\nIn the outer loop, $\\theta^*$ is optimized by taking a small step, with a linearly decreasing schedule, towards the optimized weights of the inner-loop. This allows $\\theta^*$ to converge to a nearby point (in the euclidean sense) to the optimal set of weights of each training condition. We detail the algorithm in Algorithm~\\ref{algo:meta_training}.\n\n\n\\subsubsection{Meta-adaptation}\nAt each time step, we select the most likely training condition using the previous $\\mathrm{K}$ time steps, each condition latent vector and the set of weights $\\theta^*$. \nWe then optimize the corresponding latent vector and the set of model weights, starting from $\\theta^*$, using the same optimization procedure as the inner loop but with the past $\\mathrm{K}$ steps. We detail the algorithm in Algorithm~\\ref{algo:meta_adaptation}.\n\nAfter the set of weights and the condition latent vector are optimized for the current condition, we use the MPC to select the optimal action to apply, and then new state information is collected, and the whole meta-adaptation iterates. This procedure allows any changes in the condition to be detected, and therefore the agent can adapt accordingly. \n\n\\subsection{Limitations}\nApart from the standard classical robot control of tuning PD gains and joints limits, the proposed method still requires fine-tuning of reward function, model architecture, hyper-parameters for the meta-learning and the adaptation. The use of MPC instead of a neural network policy has a trade-off between real-time computation and performance, i.e. MPC performs better in terms of adaption but requires more computation needed from the sampling procedure. \n\n\n\\begin{figure}[t]\n    \\centering\n      \\begin{subfigure}{\\linewidth}\n      \\centering\n      \\includegraphics[width=\\linewidth]{force_snapshots.png}\n      \\caption{}\n    \\end{subfigure}\n    \\begin{subfigure}{\\linewidth}\n      \\centering\n      \\includegraphics[width=\\linewidth]{slip_snapshots.png}\n      \\caption{}\n    \\end{subfigure}\n        \\begin{subfigure}{\\linewidth}\n      \\centering\n      \\includegraphics[width=\\linewidth]{damage_snapshots.png}\n      \\caption{}\n    \\end{subfigure}\n    \\begin{subfigure}{\\linewidth}\n      \\centering\n      \\includegraphics[width=\\linewidth]{tripod_snapshots.png}\n      \\caption{}\n    \\end{subfigure}\n    \\caption{Walking in presence of large uncertainties: (a) constant external force disturbance, (b) low-friction slippery ground, (c) faulty motors, and (d) with one missing leg.}\n    \n    \\vspace{-0mm}\n    \\label{fig:snapshots}\n\\end{figure}\n\n\n\\section{Results}\n\\label{sec:exp}\nWe used a custom version of the robot model (adapted from the open source SpotMicro robot~\\cite{kim_2019}) in PyBullet simulation to validate our method. Here, we first present the learning capability of the model-based RL algorithm on SpotMicro with a first adaption to a sequence of different conditions (Section~\\ref{sec:general_results}). We further validate the adaptation capability of the proposed meta-learning under various fixed frictions (Section~\\ref{sec:fixed_friction}) and time-varying, decreasing friction (Section~\\ref{sec:decreasing_friction}). \n\n\n\\subsection{Overview}\n\\label{sec:general_results}\n\n\\begin{figure}[t]\n    \\centering\n      \\begin{subfigure}{\\linewidth}\n      \\centering\n      \\includegraphics[width=\\linewidth]{velocity_snapshots_forward.png}\n      \\caption{}\n    \\end{subfigure}\n    \\begin{subfigure}{\\linewidth}\n      \\centering\n      \\includegraphics[width=\\linewidth]{velocity_snapshots_static.png}\n      \\caption{}\n    \\end{subfigure}\n        \\begin{subfigure}{\\linewidth}\n      \\centering\n      \\includegraphics[width=\\linewidth]{velocity_snapshots_backward.png}\n      \\caption{}\n    \\end{subfigure}\n    \\caption{Walking with a continuously changing velocity from 0.5m\/s to -0.2m\/s: (a) forward, (b) static, and (c) backward.}\n    \\vspace{-0mm}\n    \\label{fig:varying_velocity_snapshots}\n\\end{figure}\n\n\n\\begin{figure*}[t]\n    \\centering\n    \\includegraphics[width=\\linewidth]{only_left_side_joints.png} \n    \\caption{Measured joints trajectories of SpotMicro from a 20s test scenario where the robot or environment changed every 5s: default friction $\\mu=0.8$ (\\textcolor{YellowGreen}{\\textbf{green}}), blocked right hip pitch joint (\\textcolor{Lavender}{\\textbf{red}}), slippery ground (\\textcolor{Dandelion}{\\textbf{yellow}}) and constant external push (\\textcolor{Bittersweet}{\\textbf{brown}}).}\n    \\vspace{-2mm}\n    \\label{fig:joints_trajectories}\n\\end{figure*}\n\n\n\\begin{table*}[t]\n\\centering\n\\begin{tabular}{|l|c|c|c|c|}\n\\hline\nExpert\\textbackslash{}Condition & Default           & Slippery         & Lateral Force     & Damaged Motor      \\\\ \\hline\nDefault                         & 100\\%, 3.2$\\pm$ 0.2 & 20\\%, 0.7$\\pm$ 0.7 & 40\\%, 2.0$\\pm$ 1.0  & 100\\%, 0.7$\\pm$ 0.2  \\\\ \\hline\nSlippery                        & 30\\%, 1.3$\\pm$ 0.5  & 90\\%, 2.4$\\pm$ 0.4 & 10\\%, 0.7$\\pm$ 0.4  & 100\\%, 0.4$\\pm$  0.1 \\\\ \\hline\nLateral Force                   & 0\\%                & 0\\%                & 70\\%, 2.3 $\\pm$ 0.7 & 90\\%, 0.4$\\pm$ 0.1   \\\\ \\hline\nDamage Motor                    & 90\\%, 0.5$\\pm$ 0.2  & 70\\%, 0.3$\\pm$ 0.2 & 30\\%, 0.6$\\pm$ 0.3  & 100\\%, 2.5$\\pm$ 0.1  \\\\ \\hline\nMeta-Trained                    & 70\\%, 3.0$\\pm$ 0.6  & 80\\%, 2.4$\\pm$ 0.7 & 40\\%, 2.0$\\pm$ 1.3  & 100\\%, 2.7$\\pm$ 0.1  \\\\ \\hline\n\\end{tabular}\n\\caption{Success rate and average distance of travel for 10 episodes from different expert and the meta-trained model.}\n\\label{tab:success_rates}\n\\end{table*}\n\nWe trained the expert model with a default condition of friction $\\mu=0.8$, and this resulted default controller for walking is robust to perturbations, withstanding several pushes of 10N for 0.2s. After 300 of 10s-episodes which produced training data in the given condition, the quadruped was able to walk on slippery ground (friction coefficient $\\mu = 0.2$), against external forces or with a blocked motor or a missing\/amputated leg. Tab.~\\ref{tab:success_rates} shows the comparison between experts and meta-trained models under different conditions.\n\nUsing the proposed meta-learning method, the agent was able to adapt to four different conditions: default, fixed front-right hip motor, slippery ground and external forces. It traveled an average distance of $5.1\\pm1.5$m compared to a default expert which traveled $2.6\\pm0.8$m (averaged over 20 episodes). The joint trajectories from these test scenarios are shown in Fig.~\\ref{fig:joints_trajectories}.\n\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\linewidth]{varying_velocity.png}\n    \\caption{Online tracking of continuous and variable walking velocity using the model trained only at the fixed velocity of 0.5m\/s, while the desired velocity in the reward function changes from from 0.5m\/s to -0.2m\/s during the test scenario.}\n    \\vspace{-2mm}\n    \\label{fig:varying_velocity}\n\\end{figure}\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\linewidth]{random_velocity.png}\n    \\vspace{-4mm}\n    \\caption{Online tracking of discrete and variable walking velocity using the model trained only at the fixed velocity of 0.5m\/s, while every 2 s the desired velocity is randomly sampled within 0.5m\/s and -0.2m\/s during the test scenario.}\n    \\vspace{-0mm}\n    \\label{fig:random_velocity}\n\\end{figure}\n\n\nThe controller can achieve variable walking speed, despite being trained with only at a constant desired forward velocity, we can command different desired velocities online continuously, as shown in Fig.~\\ref{fig:varying_velocity_snapshots} and Fig.~\\ref{fig:varying_velocity}. Moreover, the trained expert controller can also generate continuous control actions to track discrete, discontinuous commanded velocities (see Fig. \\ref{fig:random_velocity}). \n\n\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\linewidth]{All_on_each_friction.png}\n    \\vspace{-0mm}\n    \\caption{Distance traveled for the full range of frictions from 3 expert models and the meta-trained model.}\n    \\vspace{-2mm}\n    \\label{fig:fixed_friction_adaptation}\n\\end{figure}\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\linewidth]{changing_friction_experts_VS_meta-learning_0.04s.png}\n    \\vspace{-0mm}\n    \\caption{Distance traveled with decreasing friction from the 3 expert models and the meta-trained model (p-value: $*<0.05$, $**<0.01$ and $***<0.001$). }\n    \\vspace{-2mm}\n    \\label{fig:changing_friction_adaptation}\n\\end{figure}\n\n\n\\subsection{Ground with Constant Friction}\n\\label{sec:fixed_friction}\nWe evaluated the adaptation capability using the meta-trained model and compared it to experts over the full range of different frictions (0.1 to 0.8 with 0.05 increments). We first trained 5 sets of experts for frictions 0.2, 0.4 and 0.6, using 300 10s-episodes, i.e., 50 minutes of data. Then we meta-trained 5 meta-models to adapt to these 3 frictions, each using one set of experts data, with the purpose that they could adapt to the full range afterwards.\n\nEach set of these 5 models was evaluated for each friction with 4 10s-episodes, so this gives 20 evaluations per expert and 20 evaluations for the meta-learning. The meta-trained models outperformed the experts on the full range of frictions, see Fig.~\\ref{fig:fixed_friction_adaptation}. As expected, each expert had its best performance when the friction constant is around its trained value.  \n\n\n\\subsection{Ground with Decreasing Friction}\n\\label{sec:decreasing_friction}j\n\nAs a comparison, we used the same experts and meta-model and benchmarked thei adaptation capability on a ground with continuously decreasing friction. We evaluated each set of models with 4 10s-episodes where friction coefficient $\\mu$ started at 0.8 and linearly decreased to 0.1. This gave 20 evaluations per expert and 20 evaluations for meta-learning.\n\nThe meta-trained models demonstrated better walking performance and traversed farther (3.38m) than the experts (3.07m), using a t-test with a p-value under $0.05$, see Fig.~\\ref{fig:changing_friction_adaptation}. In Fig.~\\ref{fig:velocity}, the curves and shaded areas are the means and standard deviations of the velocity, respectively. Snapshots of the walking gait using the meta-trained model are shown in Fig.~\\ref{fig:snapshots}, more details of walking performance can be seen in \\href{https:\/\/youtu.be\/s9OWhxorVmc}{the video here}. \n\n\nAdditionally, Fig.~\\ref{fig:condition_estimation_changing_friction} depicts the estimated condition at each time step using the past 0.1s (i.e. 5 time steps). At the beginning of the episode, when friction was higher, the model estimated a friction of 0.6 to be more likely (from 0.8 down to 0.5, i.e., 0-4.5s), then switched to a friction of 0.4 (from 0.5 down to 0.3, i.e., 4.5-7s), and finished by estimating a more likely friction of 0.2 (from 0.3 down to 0.1, i.e., 7-10s).\n\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\linewidth]{velocity_decreasing_frictions_all.png}\n    \\vspace{-4mm}\n    \\caption{Forward walking velocity with decreasing friction from 3 expert models and the meta-trained model.}\n    \\vspace{-0mm}\n    \\label{fig:velocity}\n\\end{figure}\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\linewidth]{likelyhood_estimation_0.1s.png}\n    \\vspace{-4mm}\n    \\caption{Likelihood of each pre-trained condition using online meta-adaptation in presence of decreasing friction.}\n    \\vspace{-0mm}\n    \\label{fig:condition_estimation_changing_friction}\n\\end{figure}\n\n\\subsection{Discussion}\nThe results validate the efficiency and effectiveness of the meta-learning method to detect the most probable current condition and adapt accordingly to different ground friction coefficients. The online adaptive walking using the proposed meta-learning outperformed the specialized experts which were specifically trained on specific frictions. This demonstrated the capability of meta-learning to incorporate knowledge from all training data.\n\nWe have also pushed the extreme test case in terms of unseen hardware failures. We specifically designed a case for meta-training where a motor of one leg of the quadruped was blocked at a fixed joint position (emulated actuator failures), and tested if meta-learning can adapt to the damage from a different leg faster than learning from scratch. Our investigation showed that the meta-adaptation is not able to adapt to such changes on a different leg, and we hypothesize that a second-order meta-learning algorithm, e.g. Model-Agnostic Meta-Learning~\\cite{MAML}, may be a better solution for such an extreme case.\n\n\\section{Conclusion and future work}\n\\label{sec:Conclusion}\nBased on the past work of model-based meta-RL~\\cite{FAMLE}, we have made contributions to improve the algorithm for adaptive and robust quadruped locomotion in changing robot dynamics (motor failure and amputation) and varying environmental constraints (time-varying friction, external pushes). In physics-based simulation, we have demonstrated this method can learn quadrupedal walking without using a periodic gait signal \\cite{FAMLE} or a phase vector \\cite{yang2020learning}. Instead, by updating an interaction model of the robot and environment and applying the optimal control actions, a walking gait is naturally generated as the outcome of maximizing the task reward. We further validated the capability of our proposed framework in adapting to different conditions such as robot damage, changing friction, and external force disturbances. \n\nFuture work will apply this method on the real SpotMicro robot, and identify potential issues of sim2real transfer which will be addressed by new solutions for meta-model and more effective search of model predictive control procedure. We hypothesize the current meta-learning algorithm is efficient enough for multi-task learning, which can be further studied. Also, a second-order meta-learning algorithm~\\cite{MAML} can be Incorporated could potentially achieve better adaptation to novel situations. \n\n\n\\section{Acknowledgement}\nThis work has been supported by EPSRC UK Robotics and Artificial Intelligence Hub for Offshore Energy Asset Integrity Management (EP\/R026173\/1).\n\n\n\\bibliographystyle{IEEEtran}\n\\balance\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.48\\textwidth]{figure\/Task-v0.1.pdf}\n    \\caption{\\textbf{Solving inductive reasoning tasks grounded in perception.} We consider tasks as inferring a simple formula with noisy perceptual inputs such as images. Solving this problem involves jointly learning to perceive and parse input into symbolic form (e.g., 3 numbers), and reason inductively  to recover programs that explain each arithmetic task.}\n    \\label{fig:setting}\n\\end{figure}\n\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width=0.49\\textwidth]{figure\/framework-v2.3.pdf}\n    \\caption{\\textbf{The framework of our model (ROAP) }for the neurosymbolic program synthesis problem. Given I\/O pairs $\\{(x, y)\\}$ for each task, \\textbf{\\rom{1}.}~our model first uses the symbol grounder $g$ to learn symbolic representations $s$ for high-dimensional perceptual input $x$. \\textbf{\\rom{2}.}~The program synthesizer $r$ then uses the symbol-output pairs as a program specification to infer the program for the task, $q(z| r(\\{s, y\\}))$. \\textbf{\\rom{3}.}~The differentiable interpreter finally executes the program on grounded symbols to predict the outputs $\\hat y = \\denotation{z}(s)$. \\textbf{\\rom{4}.}~The whole model is trained end-to-end using a reconstruction loss of outputs and a program length regularizer.} \n    \\label{fig:framework}\n\\end{figure}\n\nWe seek steps toward AI systems that learn to symbolically process perceptual input.\nConsider, for example, a system which learns to infer the 3D structure of objects: starting from pixels, it must infer low-level  symbols (curves, parts), and then organize them according to symbolic relationships (symmetry, part repetitions, part hierarchy).\nOr, consider a system which learns to control a moving object that navigates around obstacles: starting from sensory data (lidar, RGBD), it must first parse the world (into objects, proximities, freespace), and then compute trajectories using high-level computations (PID controllers, etc.).\nSimilar perceptual-symbolic problems arise when e.g. learning structured world models from pixels, or inferring instructions from natural language.\nWe frame such tasks as \\textbf{neurosymbolic program synthesis}: learning neural components that extract symbols from perception, and synthesizing programs to further process those symbols with more complex computations.\n\nIn this work, we study neurosymbolic program synthesis on a small-scale domain involving both perception and programmatic reasoning. Our ultimate goal is to develop general methods that could, we hope, apply to challenging neurosymbolic tasks like those previously mentioned.\nWe take the stance that symbols should be grounded in perception, and that symbol processing should be implemented by learnable program-like representations.\nHowever, we also propose that rather than hand-code a preordained set of primitive symbols, AI systems should autonomously learn to carve the perceptual world into their own discretization.\nWhat constitutes a `symbol' may vary across domain and across datasets, and can be hard for human engineers to anticipate. By jointly learning the symbols, as well as synthesizing the programs that operate on them, we hope to side-step the pitfalls associated with hand-engineered representations.\n\nConcretely, we consider regression problems mapping perceptual input $x$ to output $y$.\nThe input $x$ is first processed by a `symbol grounder' neural net $g(\\cdot )$, which produces intermediate symbolic output $s=g(x)$.\nThe symbols in $s$ are then transformed by a synthesized program $z$ in order to predict the output $\\hat{y}=\\denotation{z}(s)$. We learn $g(\\cdot )$ and $z$ by encouraging the predicted $\\hat{y}$ to be close to the ground-truth output $y$.\n\nThis setup serves as a basic model of the kinds of neurosymbolic program synthesis we seek to target.\nAlthough simple to write down, the setup is difficult to train:\nthe grounder $g(\\cdot )$ is a neural network (with unknown weights), \nwhile the program $z$ is a discrete program (with unknown structure).\nTo the best of our knowledge, there do not exist stable and scalable training procedures for such mixed discrete\/continuous search problems, especially when $g(\\cdot )$ has many parameters, and when $z$ comes from a combinatorially large space of possible programs.\nAt its core, we have a chicken-and-egg problem: if we had the program, then we could train $g(\\cdot )$ with gradient descent to predict an element of $\\denotation{z}^{-1}(y)$;\nif we had the neural net, we could synthesize the program $z$ using a variety of prior  search methods~\\cite{shah2020learning, 10.1145\/2491956.2462174} on the input-output example $g(x)\\mapsto y$.\n\nBroadly speaking, prior works have taken three approaches for addressing this mixed discrete\/continuous problem.\nFirst, the most popular strategy is to relax the discrete space of programs $z$ and train the entire system end-to-end through backpropagation~\\cite{evans2018learning, graves2014neural}.\nSecond, the complement of that approach has also been tried: discretizing the weight-space of the $g$ network and solving for both the program and the network weights using SAT~\\cite{EVANS2021103521}.\nLast, alternating optimization frameworks which successively update continuous weights and discrete structure, in the style of Expectation-Maximization, have been tried~\\cite{mws}.\n\nWe conjecture that relaxation and end-to-end backprop is likely to yield the best results, at least in the near future, because it allows one to build on the success and infrastructure of deep learning.\nTo that end, we combine three techniques to ameliorate the optimization difficulties associated with mixed discrete\/continuous problems, at least for this setting:\\\\\n\\textbf{1. Multitasking with amortized inference:} We consider a multitask setting with hundreds of interrelated programs sharing the same symbol grounding. Because they share the same symbol grounding, learning $g$ is easier, as the many tasks `triangulate' a good symbolic basis. Because the tasks share structure, we can make discrete program synthesis for $z$ tractable by \\emph{amortizing}~\\cite{gershman2014amortized} program search by training an auxiliary recognition (inference) model.\nThis `recognition model' learns to predict programs conditioned on input-output specifications.\\\\\n\\textbf{2. Overparameterization:} We `overparameterize' the space of programs: for example, even if we only expect a 4-line program, we search over the space of 20-line programs. Although this would be problematic for combinatorial search, gradient descent is well-known to benefit from overparameterized optimization landscapes~\\cite{frankle2018lottery}, and this benefit carries over to continuous relaxations of program spaces, as first noted by terpret~\\cite{gaunt2016terpret}. \\\\\n\\textbf{3. Regularization:} We encourage shorter program solutions by designing program-length regularizer compatible with continuous program relaxations. We find this aids convergence. It also combats the code explosion that overparameterization naively leads to, giving shorter and more interpretable programs.\n\nOur contribution is the combination of these techniques--\\textbf{R}egularize, \\textbf{O}verparameterize, \\textbf{A}mortize, for \\textbf{P}rograms--and so we call our approach ROAP (pronounced rope).\n\n\n\n\n\\section{Related Work}\nNeurosymbolic programming is a growing area that seeks to engineer learning and inference methods for hybrid program\/neural architectures~\\cite{PGL-049}, and our work is a special case of this broad framework.\nSpecifically, we tackle inductive program synthesis~\\cite{gulwani2017program}--synthesizing programs from input-output examples--but where the inputs are high-dimensional and must be preprocessed by a neural network into symbolic form.\nPrior works in this setting assume a hand-engineered inventory of basic symbols~\\cite{ellis2017learning}, while others backpropagate through differentiable programs to jointly train network weights and program structure~\\cite{ gaunt2017differentiable}.\nMultitasking is a known strategy for this setting~\\cite{valkov2018houdini}.\nTraining a neural network to help guide search for discrete programs (amortized inference) is standard~\\cite{devlin2017robustfill, 10.1145\/3453483.3454080}, and we extend that idea to continuous relaxations of program spaces.\n\nThe difficulties of gradient descent over relaxed program spaces is well known, to the extent that it has been dubbed the so-called `terpret problem'~\\cite{gaunt2016terpret}.\nUnfortunately, such an approach is the most straightforward way of training neural net$\\to$program architectures. From a technical perspective, our work hopes to make progress on the `terpret problem', thereby unlocking scalable and reliable training of this class of neurosymbolic programs.\n\n\n\\section{Methods}\n\\label{sec: methods}\n\\subsection{Relaxing spaces of programs}\nWe consider straight-line code: sequences of variable assignments, each of which introduces a new variable and computes its value by applying an operator to variables that were previously computed on earlier lines.\nWe write $z$ to mean the syntactic structure of a straight-line program, which is represented by a triple $(z^O, z^L, z^R)$ of binary 2D arrays encoding the syntactic structure of the program.\nGiven this discrete \\emph{sketch}~\\cite{solar2006combinatorial} of possible programs, we define a denotation operator $\\denotation{z}$ (Figure~\\ref{denotation}).\nGiven input-output examples $x\\mapsto y$, \nsingle-task neurosymbolic synthesis corresponds to\n\\begin{align}\ng(\\cdot ), z&=\\argmin_{\\substack{g(\\cdot ), z\\\\z\\text{'s entries }\\in \\left\\{ 0, 1 \\right\\}}}%\n\\left( y-\\denotation{z}(g(x)) \\right)^2\\label{map}\n\\end{align}\nThe constraint that the entries of $z$ lie in $\\left\\{ 0, 1 \\right\\}$ precludes end-to-end backpropagation.\nBecause our denotation operator is actually a smooth function of $z$, we can relax the above optimization problem by saying the entries of $z$ lie in $[ 0, 1 ]$:\n\\begin{align}\ng(\\cdot ), z&=\\argmin_{\\substack{g(\\cdot ), z\\\\z\\text{'s entries }\\in \\left[ 0, 1 \\right]}}%\n\\left( y-\\denotation{z}(g(x)) \\right)^2\\label{badrelax}\n\\end{align}\nUnfortunately the above objective is problematic: generically there is no reason for gradient descent to converge to a discrete $z$.\nWhile one can incorporate sparsity-favoring regularizers on $z$, doing so causes the probability of convergence to fall off rapidly as the target program length increases~\\cite{evans2018learning, gaunt2016terpret}, suggesting that $z$ only converges to a discrete program if it is randomly initialized close to a correct solution.\n\n\\paragraph{Overparameterization} With this setup so far, we can compactly describe the first of our three techniques: {Overparameterization}.\nWe simply expand the number of possible lines of code in the straight-line sketch, as illustrated in Figure~\\ref{fig:interpreter}, and expect that it will help convergence of the relaxed problem by providing a benign optimization landscape~\\cite{hardt2016identity, du2019gradient}.\n\n\n\n\\begin{figure*}\n\\begin{align*}\n\\denotation{z}(s)&=\\text{Exec}(z, s, L+V)\\text{\\phantom{test}\\emph{execute program and extract output on line }}L+V\\\\\n\\text{Exec}(z, s, l)&=s_l\\text{, whenever }l\\leq V\\text{\\phantom{test}\\emph{load symbolic variables as first lines of code. We have V variables}}\\\\\n\\text{Exec}(z, s, l)&=\\sum_o z^O_{lo}\\times F_o\\left( s, \\quad\\sum_{1\\leq a<l }z^L_{la}\\times\\text{Exec}(z, s, a), \\quad\\sum_{1\\leq b<l }z^R_{lb}\\times\\text{Exec}(z, s, b) \\right)\\text{, whenever }l> V\\\\\n&\\text{where }z\\text{ is a tuple of }(z^O, z^L, z^R)\\\\\nF_1(s, A, B)&=A+B\\text{\\phantom{test}\\emph{add}}\\\\\nF_2(s, A, B)&=A-B\\text{\\phantom{test}\\emph{subtraction}}\\\\\nF_3(s, A, B)&=A\\times B\\text{\\phantom{test}\\emph{multiplication}}\\\\\nF_4(s, A, B)&=A\\text{\\phantom{testttetst}\\emph{no-op\/skip connection}}\n\\end{align*}\n\\caption{\\textbf{Differentiable execution model} for a program sketch containing $L$ lines of code. $z$ parametrizes the program via a triple of 2-dimensional arrays $(z^O, z^L, z^R)$ containing values from 0-1. If $z^O_{lo}=1$, then line $l$ of the program computes its value by executing operator $o$.\nIf $z^L_{la}=1$, then line $l$ of the program gets its left argument for the operator from line $a$.\nIf $z^R_{lb}=1$, then line $l$ of the program gets its rights argument for the operator from line $b$.\nThe first $V$ lines of the program evaluate to variables stored in the symbolic representation, and we assume that there are $V$ such variables and $L$ lines of code that follow.}\\label{denotation}\n\\end{figure*}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.4\\textwidth]{figure\/interpreter-v2.1.pdf}\n    \\caption{\\textbf{Illustration of the differentiable execution model and overparameterization.} As depth $l$ increases, there are exponentially more ways to implement the same program using the straight-line-code execution model. }\n    \\label{fig:interpreter}\n\\end{figure}\n\n\\subsection{Probabilistic Framing}\nTo alleviate the problems of optimizing objective in Eq~\\ref{badrelax}, we lift our problem statement to an amortized multitask setting and now give a conditional variational autoencoder~\\cite{kingma2013auto} framing.\nWe treat the program $z$ as a latent variable which, given the input $x$, generates the output $y$ through $p(y|x, z)$.\nAs is standard, we introduce an auxiliary variational distribution $\\ensuremath{q}\\left( {z|y, x} \\right)$ to bound the marginal likelihood $p(y|x)$.\nBecause $\\ensuremath{q}\\left( {z|y, x} \\right)$ predicts programs conditioned on input-outputs, we can think of training \\ensuremath{q}~as amortizing the cost of predicting programs given specifications.\nMathematically we seek to maximize $p(y|x)$:\n\\begin{align}\np(y|x)&=\\sum_{z} p(y|x, z) p(z)\\nonumber \\\\\n&\\geq \\expect{\\ensuremath{q}}{\\log \\frac{p(y|x, z)p(z)}{\\ensuremath{q}(z|y, x)}}\\text{, ELBo}\\nonumber \\\\\n&= \\expect{\\ensuremath{q}}{\\log p(y|x, z)}+\\expect{\\ensuremath{q}}{\\log p( z)}+\\entropy{\\ensuremath{q}}\\nonumber \\\\\n&\\geq  \\underbrace{\\expect{\\ensuremath{q}}{\\log p(y|x, z)}}_\\text{reconstruction}+\\underbrace{\\expect{\\ensuremath{q}}{\\log p( z)}}_\\text{regularizer}\\label{variational}\n\\end{align}\nAbove we have dropped the term which encourages max-entropy variational approximations ($\\entropy{\\ensuremath{q}}$), partly for  convenience, and partly because we are concerned with maximum a posteriori (MAP) inference, for which a single point estimate of the posterior suffices.\n\nThe reconstruction term corresponds to the squared loss in Equation~\\ref{map}, given a suitable definition of $p(y|x, z)$.\\footnote{The negative log likelihood of a normal distribution corresponds to squared loss up to an additive constant}\nBecause our latent variable $z$ is discrete, we use a neural network $r(y, s)$ to predict the parameters of multinomial distributions over the entries of $z$:\n\\begin{align}\np(y|x, z)&=\\mathcal{N}\\left( y| \\mu=\\denotation{z}(s) \\right; \\sigma=\\sigma^t)\\\\\n\\ensuremath{q}{\\left( z|y, x \\right)}&=\\prod_{l}\\text{Multinomial}(z^{O}_l|r(y, s)^{O}_l)\\nonumber\\\\\\nonumber&\\phantom{\\prod_l}\\times\\text{Multinomial}(z^{R}_l|r(y, s)^{R}_l)\\\\&\\phantom{\\prod_l}\\times\\text{Multinomial}(z^{L}_l|r(y, s)^{L}_l)\\label{eq:multinomial}\n\\end{align}\nwhere $\\sigma^t$ is the standard deviation of $y$ for task $t$. Recall $s=g(x)$, and see Figure~\\ref{denotation} for the structure of $z$.\n\nWe optimize Equation~\\ref{variational} w.r.t. the parameters of $g$ and $r$, which are the only neural networks in our setup.\nWe use the reparameterization trick to obtain stable gradients through the sampling process for optimization~\\cite{kingma2013auto}. Typically, we use the gumbel softmax trick for each discrete multinomial distribution as defined in Equation~\\ref{eq:multinomial}.\n\n\nThis probabilistic framing ties together \\textbf{Amortization} (by training $\\ensuremath{q}$) and multitasking, by assuming we have a corpus of $\\left\\{ (x_n, y_n) \\right\\}$ pairs, for which we seek to maximize  $\\sum_n \\log p(y_n|x_n)$ using stochastic gradient descent.\n\\textbf{Regularization} enters through a suitable definition of the prior over programs, $p(z)$:\n\\begin{align}\np(z)&\\propto \\exp\\left( -\\lambda \\cdot \\text{length}(z) \\right)\n\\end{align}\nwhere $\\lambda$ is a regularization coefficient and $\\text{length}(z)$ is the length of the program encoded by $z$.\nProgram length can be recursively computed from $z$ in a manner analogous to how $z$ is recursively executed.\n\n\n\n\\subsubsection{Model structure of the program synthesizer}\n\\label{sec: program synthesizer}\nThe program synthesizer predicts the distribution of programs given output and grounded symbols $z\\sim q(\\cdot| r(y, s))$. One single symbol-output pair leaves much ambiguity for program inference. Therefore, we use multiple symbol-output pairs (together as a set) from the same task to predict the program distribution as $z\\sim q(\\cdot| r(\\{y^i, s^i\\})$.\nNoticing that this is a mapping from a set of vectors $\\{(y^{i}, s^i)\\}$ to one vector $z$, we adopt PointNet~\\cite{qi2017pointnet} to build the model. (The descriptions and justification about PointNet are stated in Appendix~\\ref{sec:pointnet}).\n\nOur program synthesizer model first extract features for the properties of the outputs $y^{i}$, the relationships between the $j$-th symbols $s_j^{i}$ and outputs, and the properties of the whole symbol-output pairs, using PointNets:\n\\begin{eqnarray*}\nh_{\\text{out}} & = & \\max_{i} \\text{MLP}_{\\text{out}}\\left(y^{i}\\right),\\\\\nh_{\\text{rel}}[j] & = & \\max_{i} \\text{MLP}_{\\text{rel}}\\left(\\left[y^{i}; s^{i}_j\\right]\\right), ~\\forall j \\in \\{1, 2, \\cdots, V\\},\\\\\nh_{\\text{io}} & = & \\max_{i} \\text{MLP}_{\\text{io}}\\left(\\left[y^{i}; s^{i}\\right]\\right),\n\\end{eqnarray*}\nwhere MLP$_\\text{out}$, MLP$_{\\text{rel}}$, and MLP$_{\\text{io}}$ are three multi-layer perceptron networks, $\\left[~\\cdot~;~\\cdot~\\right]$ means concatenation of the components, $V$ is the dimensionality of the symbolic representations $s$, and $s^{i}_j$ is the $j$-th dimensional element of $s^{i}$.\n\nThe features are then concatenated together to predict the distribution of the programs $z$ as:\n\\begin{equation*}\n    z \\sim \n    \\ensuremath{q}\\left(\\cdot| \\text{MLP}_\\text{prog}\\left(\\left[h_\\text{out}; h_\\text{io}; h_\\text{rel}[0]; h_\\text{rel}[1]; \\cdots; h_\\text{rel}[V]\\right]\\right)\\right),\n\\end{equation*}\nwhere MLP$_\\text{prog}$ is another MLP network for predicting the program distributions according to the features.\n\nIntuitively, our models can learn informative hints for synthesizing programs, such as the outputs $y^{i}$ are never prime, the outputs $y^{i}$ are always bigger than the second symbols $s^{i}_2$, and the outputs $y^{i}$ are always equal to the summation of all symbols $\\sum_{j=1}^V s^{i}_j$.\n\n\\section{Experimental Results}\nWe seek to answer the following questions in our experiments: (1) Is it feasible to jointly learn the symbolic representations from high-dimensional perceptual inputs and synthesize the programs that operate on them (Sec~\\ref{sec:exp-results-main})?  (2) Are the techniques we introduce beneficial (Sec~\\ref{sec:exp-results-ablation})? Specifically, what are the effects of multitasking with amortized inference, overparametrization, and regularization? (3) Can the generalizability of models be improved by learning interpretable symbolic representations with the programmatic prior (Sec~\\ref{sec:exp-results-generalize})?\n\n\\subsection{Experimental Setup}\n\\paragraph{Task and Dataset} As illustrated in Figure~\\ref{fig:setting}, in the neurosymbolic program synthesis problem, we are given dataset containing samples as  I\/O pairs, $\\left(x^{t,i}_1, x^{t,i}_2, \\cdots, x^{t,i}_V, y^{t,i} \\right)$, where $t, i$ index tasks and samples within tasks, respectively, and we assume the I\/O pairs are generated using gold standard symbolic representations, $s^{*,t,i}_j = g^*(x^{t,i}_j)$, and gold standard programs\n$z^{*, t}$ as \n$y^{t,i}=\\llbracket z^{*,t}\\rrbracket\\left(s^{*,t,i}_1, s^{*,t,i}_2, \\cdots, s^{*,t,i}_V\\right)$.\nWe want to train models to jointly infer the symbolic representations $s^{t,i}_j=g(x^{t,i}_j)$ and the program $\\hat z^t=\\arg\\max_z$ $q(z|r(\\{s^{t,i}_j, y^{t,i}_j\\})$ only given distant supervision via the I\/O pairs.\nSuccess means that the symbolic representations are close to the gold standard ones, $s^{t,i}_j\\simeq s^{*,t,i}_j$, and the synthesized programs are equivalent to the gold standard ones, $\\llbracket \\hat z^t\\rrbracket(s) = \\llbracket z^{*,t}\\rrbracket(s), \\forall s\\in \\mathbb{R}^V$, as shown in Figure~\\ref{fig:framework}. \n\n\n\nWe build the dataset by firstly drawing gold standard symbols, $s_j^{*, \\cdot, i}$, uniformly from $\\{1, 2, \\cdots, 10\\}$ for all $i\\le N$ and $j\\le V$, where $N$ is the number of I\/O pairs for each task and $V=3$ is the number of symbols for each input. The placeholder $\\cdot$ denotes that we share the same set of symbols $s_j^{*, \\cdot, i}$ and later inputs $x_j^{\\cdot, i}$ for all tasks $t$, for computational efficiency issues. The perceptual inputs $x_j^{\\cdot, i}$ are then drawn randomly using the CIFAR-10 dataset~\\cite{krizhevsky2009learning}. Specifically, CIFAR-10 dataset contains images in 10 classes, $\\{(\\text{img}^{k}, l^{k})\\}$, where $\\text{img}^{k}\\in \\mathbb{R}^{32\\times 32\\times 3}$ are images and $l^{k}\\in\\{1, 2, \\cdots, 10\\}$ are their labels. The perceptual inputs $x_j^{ \\cdot, i}$ are then drawn uniformly from all images $\\text{img}^{k}$ with label $l^{k}$ equal to $s_j^{*, \\cdot, i}$. The programs $z^{*,t}$ are drawn uniformly from all the semantically distinct arithmetic formulas with at most 3 variables and 3 operators, such as $s_1$, $s_2+s_3$, and $(s_1+s_3)\\times(s_1-s_3)$. The outputs $y^{t, i}$ are then calculated by executing programs on the corresponding inputs as $y^{t,i}=\\denotation{z^{*,t}}(s^{*,\\cdot, i})$, for all $t\\le T$, where $T=500$ is the number of tasks. \n\n\\paragraph{Evaluation Metrics} We evaluate the abilities of models on grounding symbols and synthesizing programs using the following metrics:\n\\begin{itemize}\n    \\item Program success rate: the ratio that the synthesized programs $\\hat z^t$ are equivalent to the gold standard programs $z^{*,t}$. In practice, we empirically evaluate the equivalence of programs using the samples in the dataset, i.e., $\\sum_{t\\le T} \\mathbf{1}(\\denotation{\\hat z^t}(s^{*, \\cdot, i})=y^{t,i}, \\forall i\\le N)~\/~T.$\n   \n    \\item Symbolic loss: the mean squared error (MSE) between the grounded symbols $s^{ \\cdot, i}_j$ and the gold standard symbols $s^{*, \\cdot, i}_j$, i.e., $\\frac{1}{NV}{\\sum_{i\\le N}\\sum_{j\\le V} \\left(g(x^{\\cdot, i}_j)-s^{*, \\cdot, i}_j\\right)^2}$.\n   \n    \\item Test loss: the mean squared error of applying the whole model to predict the outputs, i.e.,\\\\ $\\frac{1}{TN}{\\sum_{t\\le T}\\sum_{i\\le N} \\left(\\denotation{\\hat z^t}(g(x^{t, i}))-y^{t, i}\\right)^2}$.\n\\end{itemize}\nNote that, although we are evaluating models using the gold standard symbols $s^{*, \\cdot, (i)}_j$, they are not available during training and we must learn to ground them in order to achieve good performances on all metrics. Meanwhile, we use different images and random symbolic inputs for generating the training dataset and the testing dataset in order to evaluate models' robustness against the perceptual and sampling noises (CIFAR-10 dataset is split into 50000 samples for training and 10000 samples for testing). All the models and metrics are evaluated on the testing dataset.\n\n\\vspace{2.5pt}\nWe also introduce another metric for evaluating the improvements of models' generalizability from learning interpretable symbolic representations with the programmatic prior. In particular, for a new task $t'$ , we can directly apply our learned symbol grounder $g$ to get the symbolic representation $s^{t', i}_j$ of high-dimensional images $x^{t', i}_j$, and synthesize programs using the symbol-output pairs. To evaluate the out-of-distribution generalizability of such models, we train them on dataset with symbols smaller or equal to 5, $s^{t', i}_j\\le 5, \\forall i,j$, but test them on dataset with symbols larger than 5, $s^{t', i}_j> 5, \\forall i,j$. The evaluation metric is then\n\\begin{itemize}\n    \\item Generalization loss: the mean squared error of applying the model trained on smaller digits to predict outputs on larger digits, i.e., $\\frac{1}{N}{\\sum_{i\\le N} \\left(\\denotation{\\hat z^t'}(s^{t', i})-y^{t', i}\\right)^2}$.\n\\end{itemize}\nNotice that when given the learned symbol grounder, the neurosymbolic program synthesis problem reduces to the classical symbolic program synthesis problem. Any classical algorithms can be applied. In practice, we use the bottom-up enumerative search~\\cite{udupa2013transit} to find the program for the new task. \n\n\\paragraph{Implementation Details} The framework of our models is illustrated in Figure~\\ref{fig:framework} and described in Sec~\\ref{sec: methods}. We apply the standard 18-layer ResNet~\\cite{he2016deep}, which is a classical convolutional neural network, to build the symbol grounder\\footnote{The output dimensionality of the last layer of ResNet is changed to predict symbols instead of image labels.}. All MLPs used in the program synthesizer model have two hidden layers of size 256. Layer normalization~\\cite{ba2016layer} is applied before each activation function, ReLU, to improve the training stability~\\footnote{BatchNorm~\\cite{ioffe2015batch} can not be easily applied in the multi-tasking setting.}. The program sketch contains $L=30$ lines of code. \n\nGiven the large and complex search space of models, there are non-negligible probabilities that a run of the experiment would fail with loss=infinity and gradients equal to nan due to the numerical issues\\footnote{For example, when the symbols are predicted as 10 and the currently synthesized program is $s_0^{20}$, it is easy for the models' outputs to be overflow and fail the training}. To tackle this numerical issue, we instead squash the range of symbols from $\\{1, 2, \\cdots, 10\\}$ to $\\{0.1, 0.2, \\cdots, 1.0\\}$ by dividing by ten.\nAlso, for each model, we run the experiments for multiple times ($\\sim 3$) and report the best one. We select the best learned model using the mean square error of applying the learned model, $g(\\cdot), \\hat z$, to predict outputs in the training dataset. \n\n\nWe train models using the Adam~\\cite{kingma2014adam} optimizer with a learning rate equal to 3e-4 and $\\epsilon=$\\text{1e-5} for 20 epochs. The dataset contains 1e6 I\/O pairs for each task for training and 1000 I\/O pairs for each task for testing. The program length regularizer is not applied until epoch 10 with ratio $\\lambda=$2e-4. The temperature for gumbel softmax is set to 1 in the beginning and changed to 3 from epoch 15 to minimize the error gap from the continuous approximations of programs near the end of training.\n\n\n\\paragraph{Baselines} To analyze the effects of the programmatic prior and the other techniques as described in Section~\\ref{sec: methods}, we compare our model with the following baselines:\n\\begin{itemize}\n    \\item without program structure: These models use neural networks (typically MLPs) instead of programs to predict outputs from symbols as $\\hat y = \\text{MLP}(s)$. We implement two types of models to incorporate the task information and report the best of them: (1) disjoint MLPs for each task, i.e., $\\hat y^t=\\text{MLP}^t(s)$; and (2) one-hot embedding of the task id as features, i.e., $\\hat y^t=\\text{MLP}(\\left[ s;\\text{one-hot}(t)\\right])$.\n    \\item without amortized inference: These models drop the program synthesizer $r$ and instead take $z^t$ as free independent parameters. Therefore, there will be no information shared by amortized inference among the tasks.\n    \\item without gumbel softmax: These models replace gumbel softmax with softmax. So, there is no stochastic sampling, $z\\sim q(z|r(y,s))$, in these models.\n    \\item with Syntax-Tree: These models use the syntax tree execution model instead of straight-line code. More details of the syntax tree execution model are stated in Appendix~\\ref{sec:syntax-tree}. Briefly speaking, the syntax tree execution model has fewer optimal solutions in its solution space and, therefore, is less overparameterized.  \n    \\item with Depth-$D$: The program sketch contains $L=D$ lines of code, instead of $L=30$. Larger depths correspond to more overparameterized models.\n\\end{itemize}\n\n\\subsection{Joint Learning of Symbols and Programs}\n\\label{sec:exp-results-main}\n\nWe demonstrate that it is feasible to jointly learn interpretable symbolic representations and synthesize programs that operate on them using our model. As shown in Table~\\ref{tab:main}, our model successfully learns the symbolic representations and correctly synthesizes the programs for most tasks. The symbolic loss, 0.00086, is negligible for distinguishing symbols in the set $\\{0.1, 0.2, \\cdots, 1.0\\}$. And we correctly find the program for 459 of 500 tasks, which corresponds to the program success rate as 91.8\\%.\n\n\\begin{table*}[h]\n    \\centering\n   \n    \\begin{tabular}{|c|c|c|c|c|}\n    \\hline \n         & program success rate & symbolic loss & test loss & generalization loss\\\\\n    \\hline \n     ROAP (our model) & \\textbf{459}~\/~500 &  \\textbf{0.00086} & \\textbf{0.11} & \\textbf{0.07}\\\\\n     \\hline\n     w\/o program, i.e., CNN+MLP & ~~~0~\/~500 &  0.95 & \\textbf{0.11} & 1.03 \\\\\n     w\/o amortized inference & 136~\/~500 &  0.059 & 0.20 & 0.19\\\\\n     w\/o gumbel-softmax & ~~~8~\/~500 &  0.52 & 0.33 & 4.40 \\\\\n     w\/ Syntax-Tree & 345~\/~500 &  0.04 & 0.16 & 0.22 \\\\\n     w\/ depth=10 & ~58~\/~500 & 0.90 & 0.14 & 2.12\\\\\n     w\/ depth=3 & ~56~\/~500 & 1.10 & 0.16 & 2.08 \\\\\n     \\hline\n    \\end{tabular}\n    \\caption{\\textbf{The performances of models for the neurosymbolic program synthesis problem.} Our model (ROAP) successfully learned the interpretable symbolic representations (negligible symbolic loss) and the programs that operate on them (high program success rate). The learned model can predict outputs from input images with small errors on both within-distribution test dataset (test loss) and out-of-distribution test dataset with unseen larger digits (generalization loss). All techniques, including the programmatic prior, multi-tasking with amortized inference, and overparameterization, are beneficial in terms of the joint learning performances by comparing our model with the baselines.}\n   \n    \\label{tab:main}\n\\end{table*}\n\n\\subsubsection{Ablation Studies}\n\\label{sec:exp-results-ablation}\n\nWe analyze the effectiveness of techniques by performing the ablation studies and comparing our model with baselines. The experimental results show that all of our techniques, including multi-tasking with amortized inference, overparameterization, gumbel-softmax, and the program length regularizer, improve performances of models in the neurosymbolic program synthesis problem.\n\n\\paragraph{Effectiveness of multi-tasking with amortized inference} As shown in Table~\\ref{tab:main}, amortized inference with multi-tasks largely improves the performance of program synthesis  and then the learning of interpretable symbolic representations. Without amortized inference, the model can only synthesize programs correctly for 136\/500 tasks. The symbolic loss is also non-negligible as 0.059. It is because amortized inference can share information while synthesizing programs for multiple tasks and, therefore, alleviates the difficulties of program synthesis and then joint learning. More correctly synthesized programs can drive better symbolic representation learning by triangulating a good symbolic basis. Better symbolic representations, in return, enable even better program synthesis by providing more accurate program inputs. We also compare the performances of joint learning in one task, i.e., $T=1$, and 500 tasks in Table~\\ref{tab:multi-task}. We run each experiment more than 10 times, each time with randomly drawn programs, and calculate the success rate metric as the ratio of runs that learn the symbolic representations successfully, i.e., symbolic loss $\\le$ 0.05. We also report the best symbolic loss of both settings in Table~\\ref{tab:multi-task}. The results suggest that multi-tasking can largely improves the success rate of joint learning and also their performances.\n\n\\begin{table}\n    \\centering\n    \\begin{tabular}{|c|c|c|}\n    \\hline\n         & success rate & symbolic loss \\\\\n         \\hline \n    500 tasks & \\textbf{33.3\\%} & \\textbf{0.00086} \\\\\n    1 task & 10.5\\% & 0.039 \\\\\n    \\hline \n    \\end{tabular}\n    \\caption{\\textbf{Effects of multitask learning for joint learning.}}\n    \\label{tab:multi-task}\n\\end{table}\n\n\\paragraph{Effectiveness of overparameterization} Baseline models, with Syntax-Tree, with depth=10, and with depth=3, are all less overparameterized than our model. As listed in Table~\\ref{tab:main}, our overparameterized model largely outperforms these baselines in terms of all metrics. These results suggest that overparameterization helps improve the performances of joint learning, probably by providing more global optimal solutions in the solution space and avoiding converging to local optimums. We further modify the degrees of overparameterization by changing the depth $L$ of program sketches and report their performances for tasks with different difficulties in Figure~\\ref{fig:overparameterization}. The results first show that more overparameterized models perform better than less overparameterized models. Moreover, less overparameterized models perform better on easier tasks, i.e., when the relative overparameterization degree (the size of paragram sketch versus the size of correct programs) becomes larger, which also demonstrates the effectiveness of overparameterization.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.3\\textwidth]{figure\/depth-effect.pdf}\n    \\caption{\\textbf{Overparameterization is necessary for robust program recovery.} Programs get depth corresponds to the number of possible lines of code: increasing sketch depth increases overparameterization. In each cell, we show the percentage of programs of a given size successfully synthesized}\n    \\label{fig:overparameterization}\n\\end{figure}\n\n\n\\begin{figure*}[h]\n    \\centering\n   \n   \n   \n    \\includegraphics[width=0.38\\textwidth]{figure\/reg-effect.pdf}\n    \\quad\\quad\n    \\includegraphics[width=0.38\\textwidth]{figure\/reg-effect-lengths.pdf}\n    \\caption{\\textbf{Effects of the program length prior on model convergence.} The regularizer is applied starting from epoch 10. Left: number of program synthesis tasks solved during training. Right: average program length during training. On successful runs (blue),\n    the regularizer maintains performance while dramatically decreasing program length. On `salvageable' runs (orange) the regularizer pushes the model into successful solutions, but not always (green)}\n    \\label{fig:reg-effect}\n\\end{figure*}\n\n\n\\paragraph{Effectiveness of gumbel softmax} As shown in Table~\\ref{tab:main}, models without gumbel softmax perform much worse than ours. It is because the stochastic sampling noises introduced by gumbel softmax help models to explore a larger solution space and then find a better converged point while using the local optimizers such as Adam.\n\n\n\\paragraph{Effectiveness of program length prior} We demonstrate the effects of our program length regularizer in three scenarios in Figure~\\ref{fig:reg-effect}. When the model has already learned good symbols and program synthesizers (blue lines), applying the program length prior will not hurt the models' performances while can largely improve the models' interpretability by inducing synthesizing shorter programs. The averaged length of synthesized programs reduces from 21.6 to 9.6 after applying the regularization, while the program success rate increases from 431 to 455 \/ 500. For runs with poor performances, applying the program length regularization can largely boost the performances of models by providing more programmatic prior for salvageable runs (the orange lines), but not always (the green lines). Specifically, for salvageable runs, the program success rate increases from 4 to 412 \/ 500 and the symbolic loss decreases from 1.23 to 0.066.\n\n\n\\subsection{Out-of-distribution Generalizability by Programs}\n\\label{sec:exp-results-generalize}\nWe finally demonstrate the improvements of out-of-distribution generalizability of models from the programmatic prior using the generalization loss metric. The generalization loss metric evaluates the models' performances on unseen symbols (0.6 to 1.0) that are larger than the symbols (0.1 to 0.5) used during training for a new task. As shown in Table~\\ref{tab:main}, without the programmatic prior, models cannot learn the correct symbolic representations and synthesize the correct programs even though the training and test losses are already low enough. Because of the large flexibility of neural networks, models without programmatic prior can easily overfit and achieve low training losses without learning the correct representations and programs. On the other hand, our model with the programmatic prior can successfully learn the symbolic representations and programs while still achieving the same test loss. More importantly, due to the generalizability of programs, our model can generalize to unseen domains (I\/O pairs with larger digits) and achieve generalization loss as low as 0.07 (compared with 1.03 for models without the programmatic prior).\n\n\n\n\\section{Conclusion}\nOur goal is to make progress on basic neurosymbolic problems: starting from perception, and absent symbol-level supervision, how can we discover grounded symbols and the symbolic routines which manipulate them?\nOur experiments examine a small-scale arithmetic domain, and assumed the premise that end-to-end deep learning could be made to work here.\nEven in this simplistic domain, joint perceptual learning and inductive reasoning proved difficult without additional tricks.\n\nMany directions remain open.\nOne low hanging fruit is pretraining the perceptual frontend $g(\\cdot )$ with self supervised representation learning.\nBut we see the main open question as whether a pure gradient-guided search is actually superior to a hybrid alternating maximization scheme, or a pure symbolic search.\n\nTo what extent do the ROAP tricks generalize? This is an empirical question, and we are actively exploring new application areas to answer exactly that question.\nBut we are optimistic of the general applicability of the ROAP recipe, because we believe that each trick can be motivated from first principles:\n(1) Multitask learning through amortization allows sharing of statistical strength from easy tasks to hard tasks, which should facilitate bootstrap learning.\nMultitasking also constrains the symbolic representation from different directions, `triangulating' the correct symbols for the systems to be using.\n(2) Overparameterization is necessary for reliable neural network convergence, and unsurprisingly also necessary for gradient-guided program search.\n(3) Regularizing program length has solid theoretical underpinnings~\\cite{solomonoff1964formal}.\nAlthough our main demonstration is small-scale, we hope that the ideas we pilot could apply more broadly.\n\n\\section{Broader Impact}\n\nJoint learning of symbolic representations and symbolic programs that operate on them, once achieved, can largely improve models' interpretability and generalizability due to the programmatic prior, while still maintaining scalability to handle noisy high-dimensional perceptual data using neural networks with gradient descent. Human interventions and safety constraints\/verification can be further introduced through the programmatic prior. Therefore, efforts on enabling joint learning of symbols and programs, such as ours, can lead to more usages of machine learning in high-stakes and complex environments such as the medical settings, road, and commerce, where sensory inputs are noisy and high-dimensional, and interpretability and safety are critical. Joint inference of symbols and programs can also help scientific discovery by searching for interpretable and generalizable features to explain nature. On the other hand, interpretable feature extraction and convenient human intervention can be exploited to collect and analyze personal information and for malicious rumor spreading.\n\n\n\n\n\n\\section{Introduction}\n\t\n\t\\begin{figure}\n\t\t\\centering\n\t\t\\includegraphics[width=0.48\\textwidth]{figure\/Task-v0.1.pdf}\n\t\t\\caption{\\textbf{Solving inductive reasoning tasks grounded in perception.} We consider tasks as inferring a simple formula with noisy perceptual inputs such as images. Solving this problem involves jointly learning to perceive and parse input into symbolic form (e.g., 3 numbers), and reason inductively  to recover programs that explain each arithmetic task.}\n\t\t\\label{fig:setting}\n\t\\end{figure}\n\t\n\t\n\t\\begin{figure}[h]\n\t\t\\centering\n\t\t\\includegraphics[width=0.49\\textwidth]{figure\/framework-v2.3.pdf}\n\t\t\\caption{\\textbf{The framework of our model (ROAP) }for the neurosymbolic program synthesis problem. Given I\/O pairs $\\{(x, y)\\}$ for each task, \\textbf{\\rom{1}.}~our model first uses the symbol grounder $g$ to learn symbolic representations $s$ for high-dimensional perceptual input $x$. \\textbf{\\rom{2}.}~The program synthesizer $r$ then uses the symbol-output pairs as a program specification to infer the program for the task, $q(z| r(\\{s, y\\}))$. \\textbf{\\rom{3}.}~The differentiable interpreter finally executes the program on grounded symbols to predict the outputs $\\hat y = \\denotation{z}(s)$. \\textbf{\\rom{4}.}~The whole model is trained end-to-end using a reconstruction loss of outputs and a program length regularizer.} \n\t\t\\label{fig:framework}\n\t\\end{figure}\n\t\n\tWe seek steps toward AI systems that learn to symbolically process perceptual input.\n\tConsider, for example, a system which learns to infer the 3D structure of objects: starting from pixels, it must infer low-level  symbols (curves, parts), and then organize them according to symbolic relationships (symmetry, part repetitions, part hierarchy).\n\tOr, consider a system which learns to control a moving object that navigates around obstacles: starting from sensory data (lidar, RGBD), it must first parse the world (into objects, proximities, freespace), and then compute trajectories using high-level computations (PID controllers, etc.).\n\tSimilar perceptual-symbolic problems arise when e.g. learning structured world models from pixels, or inferring instructions from natural language.\n\tWe frame such tasks as \\textbf{neurosymbolic program synthesis}: learning neural components that extract symbols from perception, and synthesizing programs to further process those symbols with more complex computations.\n\t\n\tIn this work, we study neurosymbolic program synthesis on a small-scale domain involving both perception and programmatic reasoning. Our ultimate goal is to develop general methods that could, we hope, apply to challenging neurosymbolic tasks like those previously mentioned.\n\n\tWe take the stance that symbols should be grounded in perception, and that symbol processing should be implemented by learnable program-like representations.\n\tHowever, we also propose that rather than hand-code a preordained set of primitive symbols, AI systems should autonomously learn to carve the perceptual world into their own discretization.\n\tWhat constitutes a `symbol' may vary across domain and across datasets, and can be hard for human engineers to anticipate. By jointly learning the symbols, as well as synthesizing the programs that operate on them, we hope to side-step the pitfalls associated with hand-engineered representations.\n\t\n\tConcretely, we consider regression problems mapping perceptual input $x$ to output $y$.\n\tThe input $x$ is first processed by a `symbol grounder' neural net $g(\\cdot )$, which produces intermediate symbolic output $s=g(x)$.\n\n\tThe symbols in $s$ are then transformed by a synthesized program $z$ in order to predict the output $\\hat{y}=\\denotation{z}(s)$. We learn $g(\\cdot )$ and $z$ by encouraging the predicted $\\hat{y}$ to be close to the ground-truth output $y$.\n\t\n\tThis setup serves as a basic model of the kinds of neurosymbolic program synthesis we seek to target.\n\tAlthough simple to write down, the setup is difficult to train:\n\tthe grounder $g(\\cdot )$ is a neural network (with unknown weights), \n\twhile the program $z$ is a discrete program (with unknown structure).\n\tTo the best of our knowledge, there do not exist stable and scalable training procedures for such mixed discrete\/continuous search problems, especially when $g(\\cdot )$ has many parameters, and when $z$ comes from a combinatorially large space of possible programs.\n\tAt its core, we have a chicken-and-egg problem: if we had the program, then we could train $g(\\cdot )$ with gradient descent to predict an element of $\\denotation{z}^{-1}(y)$;\n\tif we had the neural net, we could synthesize the program $z$ using a variety of prior  search methods~\\cite{shah2020learning, 10.1145\/2491956.2462174} on the input-output example $g(x)\\mapsto y$.\n\t\n\tBroadly speaking, prior works have taken three approaches for addressing this mixed discrete\/continuous problem.\n\tFirst, the most popular strategy is to relax the discrete space of programs $z$ and train the entire system end-to-end through backpropagation~\\cite{evans2018learning, graves2014neural}.\n\tSecond, the complement of that approach has also been tried: discretizing the weight-space of the $g$ network and solving for both the program and the network weights using SAT~\\cite{EVANS2021103521}.\n\tLast, alternating optimization frameworks which successively update continuous weights and discrete structure, in the style of Expectation-Maximization, have been tried~\\cite{mws}.\n\t\n\tWe conjecture that relaxation and end-to-end backprop is likely to yield the best results, at least in the near future, because it allows one to build on the success and infrastructure of deep learning.\n\tTo that end, we combine three techniques to ameliorate the optimization difficulties associated with mixed discrete\/continuous problems, at least for this setting:\\\\\n\t\\textbf{1. Multitasking with amortized inference:} We consider a multitask setting with hundreds of interrelated programs sharing the same symbol grounding. Because they share the same symbol grounding, learning $g$ is easier, as the many tasks `triangulate' a good symbolic basis. Because the tasks share structure, we can make discrete program synthesis for $z$ tractable by \\emph{amortizing}~\\cite{gershman2014amortized} program search by training an auxiliary recognition (inference) model.\n\tThis `recognition model' learns to predict programs conditioned on input-output specifications.\\\\\n\t\\textbf{2. Overparameterization:} We `overparameterize' the space of programs: for example, even if we only expect a 4-line program, we search over the space of 20-line programs. Although this would be problematic for combinatorial search, gradient descent is well-known to benefit from overparameterized optimization landscapes~\\cite{frankle2018lottery}, and this benefit carries over to continuous relaxations of program spaces, as first noted by terpret~\\cite{gaunt2016terpret}. \\\\\n\t\\textbf{3. Regularization:} We encourage shorter program solutions by designing program-length regularizer compatible with continuous program relaxations. We find this aids convergence. It also combats the code explosion that overparameterization naively leads to, giving shorter and more interpretable programs.\n\t\n\tOur contribution is the combination of these techniques--\\textbf{R}egularize, \\textbf{O}verparameterize, \\textbf{A}mortize, for \\textbf{P}rograms--and so we call our approach ROAP (pronounced rope).\n\t\n\t\n\n\t\n\n\t\n\t\\section{Related Work}\n\tNeurosymbolic programming is a growing area that seeks to engineer learning and inference methods for hybrid program\/neural architectures~\\cite{PGL-049}, and our work is a special case of this broad framework.\n\tSpecifically, we tackle inductive program synthesis~\\cite{gulwani2017program}--synthesizing programs from input-output examples--but where the inputs are high-dimensional and must be preprocessed by a neural network into symbolic form.\n\tPrior works in this setting assume a hand-engineered inventory of basic symbols~\\cite{ellis2017learning}, while others backpropagate through differentiable programs to jointly train network weights and program structure~\\cite{ gaunt2017differentiable}.\n\tMultitasking is a known strategy for this setting~\\cite{valkov2018houdini}.\n\tTraining a neural network to help guide search for discrete programs (amortized inference) is standard~\\cite{devlin2017robustfill, 10.1145\/3453483.3454080}, and we extend that idea to continuous relaxations of program spaces. \n\t\n\tThe difficulties of gradient descent over relaxed program spaces is well known, to the extent that it has been dubbed the so-called `terpret problem'~\\cite{gaunt2016terpret}.\n\tUnfortunately, such an approach is the most straightforward way of training neural net$\\to$program architectures. From a technical perspective, our work hopes to make progress on the `terpret problem', thereby unlocking scalable and reliable training of this class of neurosymbolic programs.\n\t\n\tMore fundamentally, our efforts connect to the body of work on the `symbol grounding problem'~\\cite{harnad1990symbol}:\n\tHow does a system learn to `ground' abstract symbols (e.g., numbers) in terms of their high-dimensional perceptual counterparts (e.g., images of digits)?\n\tThis problem is especially difficult absent strong supervision on the meaning of each abstract symbol~\\cite{chang2020assessing}, and here we consider a distantly supervised setting.\n\tPrior works consider a variety of techniques to address symbol grounding~\\cite{topan2021techniques}.\n\tWe hope that the complimentary methods we introduce can help chip away at a small part of the symbol grounding problem.\n\n\t\n\t\\section{Methods}\n\t\\label{sec: methods}\n\t\\subsection{Relaxing Spaces of Programs}\n\tWe consider straight-line code: sequences of variable assignments, each of which introduces a new variable and computes its value by applying an operator to variables that were previously computed on earlier lines.\n\tWe write $z$ to mean the syntactic structure of a straight-line program, which is represented by a triple $(z^O, z^L, z^R)$ of binary 2D arrays encoding the syntactic structure of the program.\n\tGiven this discrete \\emph{sketch}~\\cite{solar2006combinatorial} of possible programs, we define a denotation operator $\\denotation{z}$ (Figure~\\ref{denotation}).\n\tGiven input-output examples $x\\mapsto y$, \n\tsingle-task neurosymbolic synthesis corresponds to\n\t\\begin{align}\n\t\tg(\\cdot ), z&=\\argmin_{\\substack{g(\\cdot ), z\\\\z\\text{'s entries }\\in \\left\\{ 0, 1 \\right\\}}}%\n\t\t\\left( y-\\denotation{z}(g(x)) \\right)^2\\label{map}\n\t\\end{align}\n\tThe constraint that the entries of $z$ lie in $\\left\\{ 0, 1 \\right\\}$ precludes end-to-end backpropagation.\n\tBecause our denotation operator is actually a smooth function of $z$, we can relax the above optimization problem by saying the entries of $z$ lie in $[ 0, 1 ]$:\n\t\\begin{align}\n\t\tg(\\cdot ), z&=\\argmin_{\\substack{g(\\cdot ), z\\\\z\\text{'s entries }\\in \\left[ 0, 1 \\right]}}%\n\t\t\\left( y-\\denotation{z}(g(x)) \\right)^2\\label{badrelax}\n\t\\end{align}\n\tUnfortunately the above objective is problematic: generically there is no reason for gradient descent to converge to a discrete $z$.\n\tWhile one can incorporate sparsity-favoring regularizers on $z$, doing so causes the probability of convergence to fall off rapidly as the target program length increases~\\cite{evans2018learning, gaunt2016terpret}, suggesting that $z$ only converges to a discrete program if it is randomly initialized close to a correct solution.\n\t\n\t\\paragraph{Overparameterization}  With this setup so far, we can compactly describe the first of our three techniques: {Overparameterization}.\n\tWe simply expand the number of possible lines of code in the straight-line sketch, as illustrated in Figure~\\ref{fig:interpreter}, and expect that it will help convergence of the relaxed problem by providing a benign optimization landscape~\\cite{hardt2016identity, du2019gradient}.\n\t\n\t\n\t\\begin{figure*}\n\t\t\\begin{align*}\n\t\t\t\\hspace{2cm} \\denotation{z}(s)&=\\text{Exec}(z, s, L+V)\\text{\\phantom{test}\\emph{execute program and extract output on line }}L+V\\\\\n\t\t\t\\text{Exec}(z, s, l)&=s_l\\text{, whenever }l\\leq V\\text{\\phantom{test}\\emph{load symbolic variables as first lines of code. We have V variables}}\\\\\n\t\t\t\\text{Exec}(z, s, l)&=\\sum_o z^O_{lo}\\times F_o\\left( s, \\quad\\sum_{1\\leq a<l }z^L_{la}\\times\\text{Exec}(z, s, a), \\quad\\sum_{1\\leq b<l }z^R_{lb}\\times\\text{Exec}(z, s, b) \\right)\\text{, whenever }l> V\\\\\n\t\t\t&\\text{where }z\\text{ is a tuple of }(z^O, z^L, z^R)\\\\\n\t\t\n\t\t\tF_1(s, A, B)&=A+B\\text{\\phantom{test}\\emph{add}}\\\\\n\t\t\tF_2(s, A, B)&=A-B\\text{\\phantom{test}\\emph{subtraction}}\\\\\n\t\t\tF_3(s, A, B)&=A\\times B\\text{\\phantom{test}\\emph{multiplication}}\\\\\n\t\t\tF_4(s, A, B)&=A\\text{\\phantom{testttetst}\\emph{no-op\/skip connection}}\n\t\t\n\t\t\n\t\t\\end{align*}\n\t\t\\caption{\\textbf{Differentiable execution model} for a program sketch containing $L$ lines of code. $z$ parametrizes the program via a triple of 2-dimensional arrays $(z^O, z^L, z^R)$ containing values from 0-1. If $z^O_{lo}=1$, then line $l$ of the program computes its value by executing operator $o$.\n\t\t\tIf $z^L_{la}=1$, then line $l$ of the program gets its left argument for the operator from line $a$.\n\t\t\tIf $z^R_{lb}=1$, then line $l$ of the program gets its rights argument for the operator from line $b$.\n\t\t\n\t\t\tThe first $V$ lines of the program evaluate to variables stored in the symbolic representation, and we assume that there are $V$ such variables and $L$ lines of code that follow. }\\label{denotation}\n\t\\end{figure*}\n\t\n\t\\begin{figure}\n\t\t\\centering\n\t\t\\includegraphics[width=0.4\\textwidth]{figure\/interpreter-v2.1.pdf}\n\t\t\\caption{\\textbf{Illustration of the differentiable execution model and overparameterization.} As depth $l$ increases, there are exponentially more ways to implement the same program using the straight-line-code execution model. }\n\t\t\\label{fig:interpreter}\n\t\\end{figure}\n\t\n\t\\subsection{Probabilistic Framing}\n\tTo alleviate the problems of optimizing objective in Eq~\\ref{badrelax}, we lift our problem statement to an amortized multitask setting and now give a conditional variational autoencoder~\\cite{kingma2013auto} framing.\n\tWe treat the program $z$ as a latent variable which, given the input $x$, generates the output $y$ through $p(y|x, z)$.\n\tAs is standard, we introduce an auxiliary variational distribution $\\ensuremath{q}\\left( {z|y, x} \\right)$ to bound the marginal likelihood $p(y|x)$.\n\tBecause $\\ensuremath{q}\\left( {z|y, x} \\right)$ predicts programs conditioned on input-outputs, we can think of training \\ensuremath{q}~as amortizing the cost of predicting programs given specifications.\n\tMathematically we seek to maximize $p(y|x)$:\n\t\\begin{align}\n\t\tp(y|x)&=\\sum_{z} p(y|x, z) p(z)\\nonumber \\\\\n\t\t&\\geq \\expect{\\ensuremath{q}}{\\log \\frac{p(y|x, z)p(z)}{\\ensuremath{q}(z|y, x)}}\\text{, ELBo}\\nonumber \\\\\n\t\t&= \\expect{\\ensuremath{q}}{\\log p(y|x, z)}+\\expect{\\ensuremath{q}}{\\log p( z)}+\\entropy{\\ensuremath{q}}\\nonumber \\\\\n\t\t&\\geq  \\underbrace{\\expect{\\ensuremath{q}}{\\log p(y|x, z)}}_\\text{reconstruction}+\\underbrace{\\expect{\\ensuremath{q}}{\\log p( z)}}_\\text{regularizer}\\label{variational}\n\t\\end{align}\n\tAbove we have dropped the term which encourages max-entropy variational approximations ($\\entropy{\\ensuremath{q}}$), partly for  convenience, and partly because we are concerned with maximum a posteriori (MAP) inference, for which a single point estimate of the posterior suffices.\n\t\n\tThe reconstruction term corresponds to the squared loss in Equation~\\ref{map}, given a suitable definition of $p(y|x, z)$.\\footnote{The negative log likelihood of a normal distribution corresponds to squared loss up to an additive constant}\n\tBecause our latent variable $z$ is discrete, we use a neural network $r(y, s)$ to predict the parameters of multinomial distributions over the entries of $z$:\n\t\\begin{align}\n\t\tp(y|x, z)&=\\mathcal{N}\\left( y| \\mu=\\denotation{z}(s) ; \\sigma=\\sigma^t\\right)\\\\\n\t\t\\ensuremath{q}{\\left( z|y, x \\right)}&=\\prod_{l}\\text{Multinomial}(z^{O}_l|r(y, s)^{O}_l)\\nonumber\\\\\\nonumber&\\phantom{\\prod_l}\\times\\text{Multinomial}(z^{R}_l|r(y, s)^{R}_l)\\\\&\\phantom{\\prod_l}\\times\\text{Multinomial}(z^{L}_l|r(y, s)^{L}_l)\\label{eq:multinomial}\n\t\\end{align}\n\twhere $\\sigma^t$ is the standard deviation of $y$ for task $t$. Recall $s=g(x)$, and see Figure~\\ref{denotation} for the structure of $z$.\n\t\n\tWe optimize Equation~\\ref{variational} w.r.t. the parameters of $g$ and $r$, which are the only neural networks in our setup.\n\tWe use the reparameterization trick to obtain stable gradients through the sampling process for optimization~\\cite{kingma2013auto}. Typically, we use the gumbel softmax trick for each discrete multinomial distribution as defined in Equation~\\ref{eq:multinomial}.\n\t\n\n\n\t\n\tThis probabilistic framing ties together \\textbf{Amortization} (by training $\\ensuremath{q}$) and multitasking, by assuming we have a corpus of $\\left\\{ (x_n, y_n) \\right\\}$ pairs, for which we seek to maximize  $\\sum_n \\log p(y_n|x_n)$ using stochastic gradient descent.\n\t\\textbf{Regularization} enters through a suitable definition of the prior over programs, $p(z)$:\n\t\\begin{align}\n\t\tp(z)&\\propto \\exp\\left( -\\lambda \\cdot \\text{length}(z) \\right)\n\t\\end{align}\n\twhere $\\lambda$ is a regularization coefficient and $\\text{length}(z)$ is the length of the program encoded by $z$.\n\tProgram length can be recursively computed from $z$ in a manner analogous to how $z$ is recursively executed.\n\t\n\t\n\t\n\t\\paragraph{Model Structure of the Program Synthesizer}\n\t\\label{sec: program synthesizer}\n\tThe program synthesizer predicts the distribution of programs given output and grounded symbols $z\\sim q(\\cdot| r(y, s))$. One single symbol-output pair leaves much ambiguity for program inference. Therefore, we use multiple symbol-output pairs (together as a set) from the same task to predict the program distribution as $z\\sim q(\\cdot| r(\\{y^i, s^i\\})$.\n\tNoticing that this is a mapping from a set of vectors $\\{(y^{i}, s^i)\\}$ to one vector $z$, we adopt PointNet~\\cite{qi2017pointnet} to build the model. (The descriptions and justification about PointNet are stated in Appendix~\\ref{sec:pointnet}).\n\t\n\tOur program synthesizer model first extract features for the properties of the outputs $y^{i}$, the relationships between the $j$-th symbols $s_j^{i}$ and outputs, and the properties of the whole symbol-output pairs, using PointNets:\n\t\\begin{eqnarray*}\n\t\th_{\\text{out}} & = & \\max_{i} \\text{MLP}_{\\text{out}}\\left(y^{i}\\right),\\\\\n\t\th_{\\text{rel}}[j] & = & \\max_{i} \\text{MLP}_{\\text{rel}}\\left(\\left[y^{i}; s^{i}_j\\right]\\right), ~\\forall j \\in \\{1, 2, \\cdots, V\\},\\\\\n\t\th_{\\text{io}} & = & \\max_{i} \\text{MLP}_{\\text{io}}\\left(\\left[y^{i}; s^{i}\\right]\\right),\n\t\\end{eqnarray*}\n\twhere MLP$_\\text{out}$, MLP$_{\\text{rel}}$, and MLP$_{\\text{io}}$ are three multi-layer perceptron networks, $\\left[~\\cdot~;~\\cdot~\\right]$ means concatenation of the components, $V$ is the dimensionality of the symbolic representations $s$, and $s^{i}_j$ is the $j$-th dimensional element of $s^{i}$.\n\t\n\tThe features are then concatenated together to predict the distribution of the programs $z$ as:\n\t\\begin{equation*}\n\t\tz \\sim \n\t\t\\ensuremath{q}\\left(\\cdot| \\text{MLP}_\\text{prog}\\left(\\left[h_\\text{out}; h_\\text{io}; h_\\text{rel}[0]; h_\\text{rel}[1]; \\cdots; h_\\text{rel}[V]\\right]\\right)\\right),\n\t\\end{equation*}\n\twhere MLP$_\\text{prog}$ is another MLP network for predicting the program distributions according to the features.\n\t\n\tIntuitively, our models can learn informative hints for synthesizing programs, such as the outputs $y^{i}$ are never prime, the outputs $y^{i}$ are always bigger than the second symbols $s^{i}_2$, and the outputs $y^{i}$ are always equal to the summation of all symbols $\\sum_{j=1}^V s^{i}_j$.\n\t\n\t\\section{Experimental Results}\n\tWe seek to answer the following questions in our experiments: (1) Is it feasible to jointly learn the symbolic representations from high-dimensional perceptual inputs and synthesize the programs that operate on them (Sec~\\ref{sec:exp-results-main})?  (2) Are the techniques we introduce beneficial (Sec~\\ref{sec:exp-results-ablation})? Specifically, what are the effects of multitasking with amortized inference, overparametrization, and regularization? (3) Can the generalizability of models be improved by learning interpretable symbolic representations with the programmatic prior (Sec~\\ref{sec:exp-results-generalize})?\n\t\n\t\\subsection{Experimental Setup}\n\t\\paragraph{Task and Dataset} As illustrated in Figure~\\ref{fig:setting}, in the neurosymbolic program synthesis problem, we are given dataset containing samples as  I\/O pairs, $\\left(x^{t,i}_1, x^{t,i}_2, \\cdots, x^{t,i}_V, y^{t,i} \\right)$, where $t, i$ index tasks and samples within tasks, respectively, and we assume the I\/O pairs are generated using gold standard symbolic representations, $s^{*,t,i}_j = g^*(x^{t,i}_j)$, and gold standard programs\n\t$z^{*, t}$ as \n\t$y^{t,i}=\\llbracket z^{*,t}\\rrbracket\\left(s^{*,t,i}_1, s^{*,t,i}_2, \\cdots, s^{*,t,i}_V\\right)$.\n\tWe want to train models to jointly infer the symbolic representations $s^{t,i}_j=g(x^{t,i}_j)$ and the program $\\hat z^t=\\arg\\max_z$ $q(z|r(\\{s^{t,i}_j, y^{t,i}_j\\})$ only given distant supervision via the I\/O pairs.\n\tSuccess means that the symbolic representations are close to the gold standard ones, $s^{t,i}_j\\simeq s^{*,t,i}_j$, and the synthesized programs are equivalent to the gold standard ones, $\\llbracket \\hat z^t\\rrbracket(s) = \\llbracket z^{*,t}\\rrbracket(s), \\forall s\\in \\mathbb{R}^V$, as shown in Figure~\\ref{fig:framework}. \n\t\n\n\t\n\n\t\n\tWe build the dataset by firstly drawing gold standard symbols, $s_j^{*, \\cdot, i}$, uniformly from $\\{1, 2, \\cdots, 10\\}$ for all $i\\le N$ and $j\\le V$, where $N$ is the number of I\/O pairs for each task and $V=3$ is the number of symbols for each input. The placeholder $\\cdot$ denotes that we share the same set of symbols $s_j^{*, \\cdot, i}$ and later inputs $x_j^{\\cdot, i}$ for all tasks $t$, for computational efficiency issues. The perceptual inputs $x_j^{\\cdot, i}$ are then drawn randomly using the CIFAR-10 dataset~\\cite{krizhevsky2009learning}. Specifically, CIFAR-10 dataset contains images in 10 classes, $\\{(\\text{img}^{k}, l^{k})\\}$, where $\\text{img}^{k}\\in \\mathbb{R}^{32\\times 32\\times 3}$ are images and $l^{k}\\in\\{1, 2, \\cdots, 10\\}$ are their labels. The perceptual inputs $x_j^{ \\cdot, i}$ are then drawn uniformly from all images $\\text{img}^{k}$ with label $l^{k}$ equal to $s_j^{*, \\cdot, i}$. The programs $z^{*,t}$ are drawn uniformly from all the semantically distinct arithmetic formulas with at most 3 variables and 3 operators, such as $s_1$, $s_2+s_3$, and $(s_1+s_3)\\times(s_1-s_3)$. The outputs $y^{t, i}$ are then calculated by executing programs on the corresponding inputs as $y^{t,i}=\\denotation{z^{*,t}}(s^{*,\\cdot, i})$, for all $t\\le T$, where $T=500$ is the number of tasks. \n\n\t\n\t\\paragraph{Evaluation Metrics} We evaluate the abilities of models on grounding symbols and synthesizing programs using the following metrics:\n\t\\begin{itemize}\n\t\t\\item Program success rate: the ratio that the synthesized programs $\\hat z^t$ are equivalent to the gold standard programs $z^{*,t}$. In practice, we empirically evaluate the equivalence of programs using the samples in the dataset, i.e., $\\sum_{t\\le T} \\mathbf{1}(\\denotation{\\hat z^t}(s^{*, \\cdot, i})=y^{t,i}, \\forall i\\le N)~\/~T.$\n\t\n\t\t\\item Symbolic loss: the mean squared error (MSE) between the grounded symbols $s^{ \\cdot, i}_j$ and the gold standard symbols $s^{*, \\cdot, i}_j$, i.e., $\\frac{1}{NV}{\\sum_{i\\le N}\\sum_{j\\le V} \\left(g(x^{\\cdot, i}_j)-s^{*, \\cdot, i}_j\\right)^2}$.\n\t\n\t\t\\item Test loss: the mean squared error of applying the whole model to predict the outputs, i.e.,\\\\ $\\frac{1}{TN}{\\sum_{t\\le T}\\sum_{i\\le N} \\left(\\denotation{\\hat z^t}(g(x^{t, i}))-y^{t, i}\\right)^2}$.\n\t\\end{itemize}\n\tNote that, although we are evaluating models using the gold standard symbols $s^{*, \\cdot, (i)}_j$, they are not available during training and we must learn to ground them in order to achieve good performances on all metrics. Meanwhile, we use different images and random symbolic inputs for generating the training dataset and the testing dataset in order to evaluate models' robustness against the perceptual and sampling noises (CIFAR-10 dataset is split into 50000 samples for training and 10000 samples for testing). All the models and metrics are evaluated on the testing dataset.\n\t\n\t\\vspace{2.5pt}\n\tWe also introduce another metric for evaluating the improvements of models' generalizability from learning interpretable symbolic representations with the programmatic prior. In particular, for a new task $t'$ , we can directly apply our learned symbol grounder $g$ to get the symbolic representation $s^{t', i}_j$ of high-dimensional images $x^{t', i}_j$, and synthesize programs using the symbol-output pairs. To evaluate the out-of-distribution generalizability of such models, we train them on dataset with symbols smaller or equal to 5, $s^{t', i}_j\\le 5, \\forall i,j$, but test them on dataset with symbols larger than 5, $s^{t', i}_j> 5, \\forall i,j$. The evaluation metric is then\n\t\\begin{itemize}\n\t\t\\item Generalization loss: the mean squared error of applying the model trained on smaller digits to predict outputs on larger digits, i.e., $\\frac{1}{N}{\\sum_{i\\le N} \\left(\\denotation{{{}\\hat{z}^t}'}(s^{t', i})-y^{t', i}\\right)^2}$.\n\t\\end{itemize}\n\tNotice that when given the learned symbol grounder, the neurosymbolic program synthesis problem reduces to the classical symbolic program synthesis problem. Any classical algorithms can be applied. In practice, we use the bottom-up enumerative search~\\cite{udupa2013transit} to find the program for the new task. \n\t\n\t\\paragraph{Implementation Details} The framework of our models is illustrated in Figure~\\ref{fig:framework} and described in Sec~\\ref{sec: methods}. We apply the standard 18-layer ResNet~\\cite{he2016deep}, which is a classical convolutional neural network, to build the symbol grounder\\footnote{The output dimensionality of the last layer of ResNet is changed to predict symbols instead of image labels.}. All MLPs used in the program synthesizer model have two hidden layers of size 256. Layer normalization~\\cite{ba2016layer} is applied before each activation function, ReLU, to improve the training stability~\\footnote{BatchNorm~\\cite{ioffe2015batch} can not be easily applied in the multi-tasking setting.}. The program sketch contains $L=30$ lines of code. \n\n\t\n\tGiven the large and complex search space of models, there are non-negligible probabilities that a run of the experiment would fail with loss=infinity and gradients equal to nan due to the numerical issues\\footnote{For example, when the symbols are predicted as 10 and the currently synthesized program is $s_0^{20}$, it is easy for the models' outputs to be overflow and fail the training}. To tackle this numerical issue, we instead squash the range of symbols from $\\{1, 2, \\cdots, 10\\}$ to $\\{0.1, 0.2, \\cdots, 1.0\\}$ by dividing by ten.\n\tAlso, for each model, we run the experiments for multiple times ($\\sim 3$) and report the best one. We select the best learned model using the mean square error of applying the learned model, $g(\\cdot), \\hat z$, to predict outputs in the training dataset. \n\n\t\n\n\t\n\tWe train models using the Adam~\\cite{kingma2014adam} optimizer with a learning rate equal to 3e-4 and $\\epsilon=$\\text{1e-5} for 20 epochs. The dataset contains 1e6 I\/O pairs for each task for training and 1000 I\/O pairs for each task for testing. The program length regularizer is not applied until epoch 10 with ratio $\\lambda=$2e-4. The temperature for gumbel softmax is set to 1 in the beginning and changed to 3 from epoch 15 to minimize the error gap from the continuous approximations of programs near the end of training.\n\t\n\t\n\t\\paragraph{Baselines} To analyze the effects of the programmatic prior and the other techniques as described in Section~\\ref{sec: methods}, we compare our model with the following baselines:\n\t\\begin{itemize}\n\t\t\\item without program structure: These models use neural networks (typically MLPs) instead of programs to predict outputs from symbols as $\\hat y = \\text{MLP}(s)$. We implement two types of models to incorporate the task information and report the best of them: (1) disjoint MLPs for each task, i.e., $\\hat y^t=\\text{MLP}^t(s)$; and (2) one-hot embedding of the task id as features, i.e., $\\hat y^t=\\text{MLP}(\\left[ s;\\text{one-hot}(t)\\right])$.\n\t\t\\item without amortized inference: These models drop the program synthesizer $r$ and instead take $z^t$ as free independent parameters. Therefore, there will be no information shared by amortized inference among the tasks.\n\t\t\\item without gumbel softmax: These models replace gumbel softmax with softmax. So, there is no stochastic sampling, $z\\sim q(z|r(y,s))$, in these models.\n\t\t\\item with Syntax-Tree: These models use the syntax tree execution model instead of straight-line code. More details of the syntax tree execution model are stated in Appendix~\\ref{sec:syntax-tree}. Briefly speaking, the syntax tree execution model has fewer optimal solutions in its solution space and, therefore, is less overparameterized.  \n\t\t\\item with Depth-$D$: The program sketch contains $L=D$ lines of code, instead of $L=30$. Larger depths correspond to more overparameterized models.\n\t\\end{itemize}\n\t\n\t\\subsection{Joint Learning of Symbols and Programs}\n\t\\label{sec:exp-results-main}\n\t\n\tWe demonstrate that it is feasible to jointly learn interpretable symbolic representations and synthesize programs that operate on them using our model. As shown in Table~\\ref{tab:main}, our model successfully learns the symbolic representations and correctly synthesizes the programs for most tasks. The symbolic loss, 0.00086, is negligible for distinguishing symbols in the set $\\{0.1, 0.2, \\cdots, 1.0\\}$. And we correctly find the program for 459 of 500 tasks, which corresponds to the program success rate as 91.8\\%.\n\t\n\t\\begin{table*}[h]\n\t\t\\centering\n\t\t\\caption{\\textbf{The performances of models for the neurosymbolic program synthesis problem.} Our model (ROAP) successfully learned the interpretable symbolic representations (negligible symbolic loss) and the programs that operate on them (high program success rate). The learned model can predict outputs from input images with small errors on both within-distribution test dataset (test loss) and out-of-distribution test dataset with unseen larger digits (generalization loss). All techniques, including the programmatic prior, multi-tasking with amortized inference, and overparameterization, are beneficial in terms of the joint learning performances by comparing our model with the baselines. }\n\t\n\t\t\\label{tab:main}\n\t\n\t\t\\begin{tabular}{|c|c|c|c|c|}\n\t\t\t\\hline \n\t\t\t& program success rate & symbolic loss & test loss & generalization loss\\\\\n\t\t\t\\hline \n\t\t\tROAP (our model) & \\textbf{459}~\/~500 &  \\textbf{0.00086} & \\textbf{0.11} & \\textbf{0.07}\\\\\n\t\t\t\\hline\n\t\t\tw\/o program, i.e., CNN+MLP & ~~~0~\/~500 &  0.95 & \\textbf{0.11} & 1.03 \\\\\n\t\t\tw\/o amortized inference & 136~\/~500 &  0.059 & 0.20 & 0.19\\\\\n\t\t\tw\/o gumbel-softmax & ~~~8~\/~500 &  0.52 & 0.33 & 4.40 \\\\\n\t\t\tw\/ Syntax-Tree & 345~\/~500 &  0.04 & 0.16 & 0.22 \\\\\n\t\t\tw\/ depth=10 & ~58~\/~500 & 0.90 & 0.14 & 2.12\\\\\n\t\t\tw\/ depth=3 & ~56~\/~500 & 1.10 & 0.16 & 2.08 \\\\\n\t\t\t\\hline\n\t\t\\end{tabular}\n\t\\end{table*}\n\t\n\t\\subsubsection{Ablation Studies}\n\t\\label{sec:exp-results-ablation}\n\t\n\tWe analyze the effectiveness of techniques by performing the ablation studies and comparing our model with baselines. The experimental results show that all of our techniques, including multi-tasking with amortized inference, overparameterization, gumbel-softmax, and the program length regularizer, improve performances of models in the neurosymbolic program synthesis problem.\n\t\n\t\\paragraph{Effectiveness of multi-tasking with amortized inference} As shown in Table~\\ref{tab:main}, amortized inference with multi-tasks largely improves the performance of program synthesis  and then the learning of interpretable symbolic representations. Without amortized inference, the model can only synthesize programs correctly for 136\/500 tasks. The symbolic loss is also non-negligible as 0.059. It is because amortized inference can share information while synthesizing programs for multiple tasks and, therefore, alleviates the difficulties of program synthesis and then joint learning. More correctly synthesized programs can drive better symbolic representation learning by triangulating a good symbolic basis. Better symbolic representations, in return, enable even better program synthesis by providing more accurate program inputs. We also compare the performances of joint learning in one task, i.e., $T=1$, and 500 tasks in Table~\\ref{tab:multi-task}. We run each experiment more than 10 times, each time with randomly drawn programs, and calculate the success rate metric as the ratio of runs that learn the symbolic representations successfully, i.e., symbolic loss $\\le$ 0.05. We also report the best symbolic loss of both settings in Table~\\ref{tab:multi-task}. The results suggest that multi-tasking can largely improves the success rate of joint learning and also their performances.\n\t\n\t\\begin{table}\n\t\t\\centering\n\t\t\\caption{\\textbf{Effects of multitask learning for joint learning.}}\n\t\t\\label{tab:multi-task}\n\t\t\\begin{tabular}{|c|c|c|}\n\t\t\t\\hline\n\t\t\t& success rate & symbolic loss \\\\\n\t\t\t\\hline \n\t\t\t500 tasks & \\textbf{33.3\\%} & \\textbf{0.00086} \\\\\n\t\t\t1 task & 10.5\\% & 0.039 \\\\\n\t\t\t\\hline \n\t\t\\end{tabular}\n\t\\end{table}\n\t\n\t\\paragraph{Effectiveness of overparameterization} Baseline models, with Syntax-Tree, with depth=10, and with depth=3, are all less overparameterized than our model. As listed in Table~\\ref{tab:main}, our overparameterized model largely outperforms these baselines in terms of all metrics. These results suggest that overparameterization helps improve the performances of joint learning, probably by providing more global optimal solutions in the solution space and avoiding converging to local optimums. We further modify the degrees of overparameterization by changing the depth $L$ of program sketches and report their performances for tasks with different difficulties in Figure~\\ref{fig:overparameterization}. The results first show that more overparameterized models perform better than less overparameterized models. Moreover, less overparameterized models perform better on easier tasks, i.e., when the relative overparameterization degree (the size of paragram sketch versus the size of correct programs) becomes larger, which also demonstrates the effectiveness of overparameterization.\n\t\n\t\\begin{figure}\n\t\t\\centering\n\t\t\\includegraphics[width=0.3\\textwidth]{figure\/depth-effect.pdf}\n\t\t\\caption{\\textbf{Overparameterization is necessary for robust program recovery.} Programs get depth corresponds to the number of possible lines of code: increasing sketch depth increases overparameterization. In each cell, we show the percentage of programs of a given size successfully synthesized}\n\t\t\\label{fig:overparameterization}\n\t\\end{figure}\n\t\n\t\n\t\\begin{figure*}[h]\n\t\t\\centering\n\t\n\t\n\t\n\t\t\\includegraphics[width=0.38\\textwidth]{figure\/reg-effect.pdf}\n\t\t\\quad\\quad\n\t\t\\includegraphics[width=0.38\\textwidth]{figure\/reg-effect-lengths.pdf}\n\t\t\\caption{\\textbf{Effects of the program length prior on model convergence.} The regularizer is applied starting from epoch 10. Left: number of program synthesis tasks solved during training. Right: average program length during training. On successful runs (blue),\n\t\t\tthe regularizer maintains performance while dramatically decreasing program length. On `salvageable' runs (orange) the regularizer pushes the model into successful solutions, but not always (green)}\n\t\t\\label{fig:reg-effect}\n\t\\end{figure*}\n\t\n\t\n\t\\paragraph{Effectiveness of gumbel softmax} As shown in Table~\\ref{tab:main}, models without gumbel softmax perform much worse than ours. It is because the stochastic sampling noises introduced by gumbel softmax help models to explore a larger solution space and then find a better converged point while using the local optimizers such as Adam.\n\t\n\t\n\t\\paragraph{Effectiveness of program length prior} We demonstrate the effects of our program length regularizer in three scenarios in Figure~\\ref{fig:reg-effect}. When the model has already learned good symbols and program synthesizers (blue lines), applying the program length prior will not hurt the models' performances while can largely improve the models' interpretability by inducing synthesizing shorter programs. The averaged length of synthesized programs reduces from 21.6 to 9.6 after applying the regularization, while the program success rate increases from 431 to 455 \/ 500. For runs with poor performances, applying the program length regularization can largely boost the performances of models by providing more programmatic prior for salvageable runs (the orange lines), but not always (the green lines). Specifically, for salvageable runs, the program success rate increases from 4 to 412 \/ 500 and the symbolic loss decreases from 1.23 to 0.066.\n\t\n\t\n\t\\subsection{Out-of-Distribution Generalizability by Programs}\n\t\\label{sec:exp-results-generalize}\n\t\n\t\n\tWe demonstrate the improvements of out-of-distribution generalizability of models from the programmatic prior using the generalization loss metric. The generalization loss metric evaluates the models' performances on unseen symbols (0.6 to 1.0) that are larger than the symbols (0.1 to 0.5) used during training for a \\emph{new} task, having already seen those symbols used in \\emph{old} tasks. As shown in Table~\\ref{tab:main}, without the programmatic prior, models cannot learn the correct symbolic representations and synthesize the correct programs even though training and test losses are low. Because of the large flexibility of neural networks, models without a programmatic prior can easily overfit and achieve low training losses without learning the correct representations and programs. On the other hand, our model with the programmatic prior can successfully learn the symbolic representations and programs while still achieving the same test loss. The symbol grounder is shared among multiple tasks so that it can learn to ground unseen symbols for the new task. More importantly, due to the generalizability of programs, our programmatic model can generalize to unseen domains (I\/O pairs with larger digits) and achieve generalization loss as low as 0.07 (compared with 1.03 for models without the programmatic prior).\n\t\n\t\n\t\\section{Conclusion}\n\tOur goal is to make progress on basic neurosymbolic problems: starting from perception, and absent symbol-level supervision, how can we discover grounded symbols and the symbolic routines which manipulate them?\n\tOur experiments examine a small-scale arithmetic domain, and assumed the premise that end-to-end deep learning could be made to work here.\n\tEven in this simplistic domain, joint perceptual learning and inductive reasoning proved difficult without additional tricks.\n\t\n\tMany directions remain open.\n\tOne low hanging fruit is pretraining the perceptual frontend $g(\\cdot )$ with self supervised representation learning.\n\tBut we see the main open question as whether a pure gradient-guided search is actually superior to a hybrid alternating maximization scheme, or a pure symbolic search.\n\t\n\tTo what extent do the ROAP tricks generalize? This is an empirical question, and we are actively exploring new application areas to answer exactly that question.\n\tBut we are optimistic of the general applicability of the ROAP recipe, because we believe that each trick can be motivated from first principles:\n\t(1) Multitask learning through amortization allows sharing of statistical strength from easy tasks to hard tasks, which should facilitate bootstrap learning.\n\tMultitasking also constrains the symbolic representation from different directions, `triangulating' the correct symbols for the systems to be using.\n\t(2) Overparameterization is necessary for reliable neural network convergence, and unsurprisingly also necessary for gradient-guided program search.\n\t(3) Regularizing program length has solid theoretical underpinnings~\\cite{solomonoff1964formal}.\n\tAlthough our main demonstration is small-scale, we hope that the ideas we pilot could apply more broadly.\n\t\n\t\\begin{figure*}[h]\n\t\t\\begin{align*}\n\t\t\t\\hspace{2.5cm}\n\t\t\t\\denotation{z}(s)&=\\text{Exec}(z, s, 0)\\text{\\phantom{test}\\emph{execute program and extract output on the root node with }}d=0\\\\\n\t\t\t\\text{Exec}(z, s, path)&=\\sum_o z^{{path}}\\times F_o\\left( s, \\quad\\text{Exec}(z, s, path.0), \\quad\\text{Exec}(z, s, path.1) \\right)\\text{, whenever len}(path)<D\\\\\n\t\t\t\\text{Exec}(z, s, path)&=\\sum_{1\\le j\\le V}z^{path}_j\\times s_j\\text{, whenever len}(path)=D \\text{\\phantom{test}\\emph{load symbolic variables in leaf nodes}}\\\\\n\t\t\t&\\text{where }z\\text{ is a tuple of }(z^0, z^{0.0}, z^{0.1}, z^{0.0.0}, z^{0.0.1}, \\cdots)\\\\\n\t\t\n\t\t\tF_1(s, A, B)&=A+B\\text{\\phantom{test}\\emph{add}}\\\\\n\t\t\tF_2(s, A, B)&=A-B\\text{\\phantom{test}\\emph{subtraction}}\\\\\n\t\t\tF_3(s, A, B)&=A\\times B\\text{\\phantom{test}\\emph{multiplication}}\\\\\n\t\t\tF_4(s, A, B)&=A\\text{\\phantom{testttetst}\\emph{no-op\/skip connection}}\n\t\t\n\t\t\n\t\t\\end{align*}\n\t\t\\caption{The syntax-tree execution model for a program sketch with a depth-$D$ syntax tree. $z$ parametrizes the program by specifying the operator for all internal nodes in the syntax tree, $z^{path}$ when len$(path)<D$, and specifying the symbolic variables for all leaf nodes, $z^{path}$ when len$(path)=D$.}\n\t\t\\label{fig:syntax-tree}\n\t\\end{figure*}\n\t\n\t\n\n\t\\section{Broader Impact}\n\t\n\tJoint learning of symbolic representations and symbolic programs that operate on them, once achieved, can largely improve models' interpretability and generalizability due to the programmatic prior, while still maintaining scalability to handle noisy high-dimensional perceptual data using neural networks with gradient descent. Human interventions and safety constraints\/verification can be further introduced through the programmatic prior. Therefore, efforts on enabling joint learning of symbols and programs, such as ours, can lead to more usages of machine learning in high-stakes and complex environments such as the medical settings, road, and commerce, where sensory inputs are noisy and high-dimensional, and interpretability and safety are critical. Joint inference of symbols and programs can also help scientific discovery by searching for interpretable and generalizable features to explain nature. On the other hand, interpretable feature extraction and convenient human intervention can be exploited to collect and analyze personal information and for malicious rumor spreading.\n\t\n\t\n\n\n\n\n\n\t\\begin{acks}\n\t\tWe are grateful for helpful discussions with Owen Lewis about overparameterization.\n\t\\end{acks}\n\t\n\t\n\t\n\n\n\t\n\t\\begin{figure}[]\n\t\t\\centering\n\t\t\\includegraphics[width=0.45\\textwidth]{figure\/syntax-tree.pdf}\n\t\t\\caption{An example of the syntax trees for $y=s_1\\times s_2 + s_3$. The symbolic variables are loaded in the leaf nodes and the operators are assigned in the internal nodes.  }\n\t\t\\label{fig:syntax-tree-example}\n\t\\end{figure}\n\t\n\t\n\t\n\t","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe past one hundred years have witnessed incredible advancements in artificial intelligence (AI)~\\cite{stone2022artificial,littman2022gathering}. These advancements are integrating cyberspace (CS), physical space (PS), and social space(SS), into the cyber-physical-social-system(CPSS), which extraordinarily expands the territories of human civilizations~\\cite{wang2010emergence} extraordinarily. However, AI \\textit{per se} is not enough to establish trust in the CPSS~\\cite{jacovi2021formalizing,jan2020ai}, which is the cornerstone of prosperity in every civilized society. Blockchain, also coined as \\textit{decentralized AI}, has the potential to empower AI to be more trustworthy by creating a decentralized trust of privacy, security, and audit-ability~\\cite{harris2019decentralized,adel2022decentralizing,hussain2021artificial}. However, systematic studies on the design principle of blockchain as a trust engine for the CPSS are still absent. If we could decipher the philosophy of blockchain, we would build the infrastructure of a better digital world. In this article, we provide an initiative for seeking the design principle of blockchain for the betterment of human civilization. Unlike the existing systemization of knowledge (SoK) on blockchain that focuses on a specific discipline purpose or topic of interest, ours aims at opening an intellectual conversation beyond boundaries for the ultimate goal of a better digital world, namely, the SoK of SoKs. Specifically, we hope to initiate the answers to three questions for the past, the current, and the future of blockchain design.\n\\begin{enumerate}\n    \\item {\\em The past:} Does blockchain live up to its original design principles as a distributed database? \n    \\item {\\em The current:} How do the current blockchain literature, industry practices, and global standards address and develop the design principle of blockchain?\n    \\item {\\em The future:} What are the gaps between the current design of blockchain and the design principle of a trust engine for a truly intelligent world?\n\\end{enumerate}\n\nIn \\Cref{sec: the past}, we examine the performance of blockchain according to its original design principle as a distributed database. We present the controversial answers in a dialog style of debate. In \\Cref{sec: the current}, we analyze the current research and practice of blockchain principles by investigating the current SoKs of emerging blockchain literature, the white papers that provide technique cornerstones for blockchain technology in the real world, and the discussions on global standards for blockchain. We identify a taxonomy of blockchain literature into the seven categories of privacy and security, scalability, decentralization, applicability, governance and regulation, system design, and cross-chain interoperability. Integrating the AI of natural language processing (NLP)~\\cite{bird2009natural} methods, we find that the current blockchain design principles, in both research and practice, are more centered around the first category of privacy and security and the fourth category of applicability. Future scholars, practitioners, and policy-makers have vast opportunities in the other much less exploited categories and the synthesis at the interface of multiple categories. In \\Cref{sec: the future}, we envision the future of blockchain technology by pointing out the gaps between the current design of blockchain and the design principle of a trust engine for a truly intelligent world. By providing counterexamples, we question the possibility of developing a plausible solution of singularity to the gaps without crossing the current boundaries of domain expertise.\n\\par\nSages in both the West and East have been craving an ideal society. In the \\textit{Book of Rites}, one of the Confucian (551\u2013479 B.C.E.) classics~\\cite{sep-confucius}, the \\textit{Great Unity} is a Chinese vision of an ideal world in which men of virtue and ability rule. In ancient Greece, Plato~\\cite{sep-plato} (429\u2013347 B.C.E.) envisioned an ideal city-state ruled by a philosopher-king of justice in the \\textit{Republic}, a Socratic dialog. By such a coincidence, both Confucius and Plato place the foundation of an ideal society on cultivating or selecting the men of virtue to be the administrators whom people can trust. Nevertheless, the existence of a benevolent ruler might not be guaranteed. In contrast, blockchain enables a new way of trust creation empowered by computer intelligence beyond human ethics. However, the current blockchain technology is evidently not a panacea for doubts about human trust. Then, why \"throw the baby out with the bathwater?\" Existing mechanisms for cultivating human trust have created economic prosperity and human civilizations for thousands of years. How then can we integrate blockchain technology into the existing governance, economics, and ethics to create an ideal society in the new era of CPSS? We initiate an open dialog of synthetic solutions. \n\\section{The past}\n\\label{sec: the past}\n\n\\begin{figure}\n  \\includegraphics[width=\\textwidth]{figs\/Sunshine_and_ChatGPT3.png}\n  \\caption{A Conversation Between Sunshine and Chat GPT-3 on 2022\/12\/30}\n  \\label{fig:chatGPT3}\n\\end{figure}\n\nWhat was blockchain originally designed for? How does blockchain live up to the design principle of its original purpose? \\citet{sherman2019origins} introduce the origin of blockchain for establishing a distributed database trusted by mutually suspicious groups. \\citet{narayanan2016bitcoin, haber1990time},and~\\citet{bayer1993improving} further elaborate on the follow-up development of blockchain technology for a series of desired properties such as decentralization, immutability, transparency, security, and efficiency. \\Cref{fig:chatGPT3} shows the answer returned by the Generative Pre-trained Transformer (ChatGPT),~\\url{https:\/\/openai.com\/blog\/chatgpt\/}, a chatbot launched by OpenAI in November 2022, to the question of \"what is the design principle of blockchain?\" The answer accurately matches elaborations provided in earlier blockchain literature. We now examine the performance of blockchain on each principle in a dialog style of debate in the spirit of Confucius's \\textit{the Analects}~\\cite{ames2010analects} and Plato's \\textit{the Republic}~\\cite{plato2005republic}. We find the answer to be controversial. Moreover, the desired principles might either conflict with each other or lead to other undesired consequences.  \n\\subsection*{Question 1: Does blockchain live up to its promise of decentralization?}\n\\subsubsection{Pros}\nYes, blockchain is decentralized. Compared with centralized ledgers, blockchain is not subject to the single point failure of a centralized entity~\\cite{puthal2018blockchain}. Blockchain records and validates data by a consensus protocol involving decentralized entities, which has the fault tolerance of minorities~\\cite{zhang2022blockchain}. \n\\subsubsection{Cons}\nNo, blockchain is not decentralized. \\citet{sai2021taxonomy} provide a taxonomy of blockchain centralization in 13 aspects. For example, \\citet{sai2021taxonomy} show that the top four mining pools in Ethereum (Bitcoin) consist of 63.04\\% (50.36\\%) of the hashing power, which is enough to conduct a successful malicious attack. Moreover, \\citet{zhang2022sok} and the references therein show that the usage of blockchain at the application layer has the trend of conversing to centralization.   \n\n\\subsection*{Question 2: Does blockchain live up to its promise of immutability?}\n\\subsubsection{Pros}\nYes, blockchain is immutable. ~\\citet{narayanan2016bitcoin} elaborate on the difficulty of manipulating recorded data due to the chain structure of Merkle trees. \n\\subsubsection{Cons}\nNo, blockchain is mutable. \\citet{hofmann2017immutability} show that the immutability of blockchain can be breached by various attacks. \\citet{politou2019blockchain} even address the conflicts between blockchain immutability and the new European data protection regulation on the right to be forgotten (RtbF), according to which individuals have the right to delete their personal data under certain conditions. \n\n\\subsection*{Question 3: Does blockchain live up to its promise of transparency?}\n\n\\subsubsection{Pros}\nYes, blockchain is transparent. ~\\cite{monrat2019survey} explains that all the transactions on blockchain are recorded and auditable by each replica node in the network. \n\\subsubsection{Cons}\nNo, blockchain is not transparent. \\citet{sai2021taxonomy} demonstrates the centralization in node operation, which is required to audit the blockchain data due to technique barriers. \\citet{feng2019survey} even point out the transparency of blockchain as a negative feature and a threat to user privacy. \n\n\\subsection*{Question 4: Does blockchain live up to its promise of security?}\n\\subsubsection{Pros}\nYes, blockchain is secure. \\citet{zhang2019security} address the blockchain security property of consistency and integrity achieved by the main techniques of consensus algorithms and hash-chained storage. \n\\subsubsection{Cons}\nNo, blockchain is not secure. \\citet{lin2017survey} evidence a long list of real attacks on blockchain together with an analysis of potential security issues. \\citet{budish2022economic} even establishes a conclusion that the mechanism supporting blockchain security is either vulnerable or not worth the economic cost. \n\n\\subsection*{Question 5: Does blockchain live up to its promise of efficiency?}\n\\subsubsection{Pros}\nYes, blockchain is efficient. \\citet{wang2019blockchain} analyze how blockchain improves efficiency in human interactions by automating and streamlining business operations using smart contracts. \n\\subsubsection{Cons}\nNo, blockchain is not efficient. Current major blockchains can only sustain tens of transactions per second (TPS), which is not comparable to the centralized platforms, which have a throughput of thousands of TPS~\\cite{chauhan2018blockchain}.\n\n\n\\section{The current}\n\\label{sec: the current}\nHow do the current blockchain literature, industry practices, and global standards address and develop the design principle of blockchain? We collect all the systemization of knowledge (SoK) papers related to facets of blockchain or surveys and review articles that satisfy the defining features of SoKs. We refer the readers to the JSys website, \\url{https:\/\/www.jsys.org\/type_SoK\/}, for background information on SoKs and the Oakland Conference GitHub site, \\url{https:\/\/oaklandsok.github.io\/}, for the histories of SoKs in computer security and privacy literature. We present a taxonomy of the collected SoKs in \\Cref{tab1} and \\Cref{tab2}, broken into seven categories based on the topic of research questions: \n\\begin{itemize}\n    \\item {\\em Category 1: Privacy and Security.} We include literature that answers questions about private information protection and attacks on blockchain systems. We refer the readers to \\citet{sep-privacy} for a philosophical symposium on \\textit{privacy} and \\citet{bishop2003computer} for the computer science insights on \\textit{security}.\n    \\item {\\em Category 2: Scalability.} We include literature that answers questions related to the scalable deployment of blockchain, usually measured in throughput and quality of service. We refer the readers to \\citet{jogalekar2000evaluating} for the definition and measurement of scalability in the distributed system literature. \n    \\item {\\em Category 3: Decentralization.} We include the literature that answers questions related to the designed and realized decentralization of blockchain. We refer the readers to \\citet{zhang2022sok} for an interdisciplinary synthesis of a taxonomy of blockchain decentralization. \n    \\item {\\em Category 4: Applicability.} We include literature that answers questions related to the application of blockchain to solve social and economic issues in various of human interactions. \\Cref{tab2} shows that the major applications are in financial services~\\cite{zetzsche2020decentralized,chen2020blockchain,harvey2021defi}. \n    \\item {\\em Category 5: Governance and Regulation.} We include literature that answers questions related to utilizing blockchain as a new way of governance, the process of interaction of an organized society, and its interactions with existing governance solutions, including corporate, government, and nongovernment organizations (NGOs). We refer the readers to \\cite{bevir2012governance} for a review of existing governance solutions, including various forms of social coordination and patterns of rules.    \n    \\item {\\em Category 6: System design.} We include literature that answers questions related to designing blockchain as an advancement of existing computer systems. We refer the readers to \\citet{saltzer2009principles} for the design principles of computer systems. \n    \\item {\\em Category 7: Cross-chain and Interoperability.} We include literature that answers questions related to communication, cooperation, and integration among blockchain systems, other computer systems, and human societies. We refer the readers to \\citet{wegner1996interoperability} and \\citet{leal2019interoperability} for the importance and assessment of interoperability among information systems. \n\\end{itemize}\n\n\\Cref{tab6} lists the top 28 blockchain projects, excluding applications on the blockchain system and cross-chain solutions ranked by the market value of their native currency retrieved from Coinmarketcap, \\url{https:\/\/coinmarketcap.com\/}, on Dec. 27, 2022. We further collect the white papers that provide the technical foundations for the 28 projects. We then conduct a text analysis applying natural language processing (NLP) methods to produce a word cloud and bigram networks of the titles and abstracts of the SoKs and white papers. The results are presented in \\Cref{tab4}, \\Cref{tab5}, \\Cref{tab7}, \\Cref{tab8}, \\Cref{tab9}, \\Cref{fig:teaser}, \\Cref{fig:whitepaper_bigram_title}, \\Cref{fig:wordcloud_abstract}, \\Cref{fig:whitepaper_wordcloud_abstract}, \\Cref{fig:whitepaper_wordcloud_title}, \\Cref{fig:bigram_title}, \\Cref{fig:bigram_abstract}, and \\Cref{fig:whitepaper_bigram_abstract}, \\Cref{fig:whitepaper_bigram_title}. The word cloud distinguishes the word frequency in these documents through font size, and the bigram represents the coappearance of two words in sequential order ranking by counts in the tables and in the network figures. We also further research the emerging documents of blockchain standard development and papers discussing blockchain standards by working groups and institutions globally.  \n\\par\n\\textbf{Data and Code Availability}: We open source the data and code for replication and future research on the GitHub: \\url{https:\/\/github.com\/sunshineluyao\/design-principle-blockchain}. \n    \nWe find that the current blockchain design principles, in both research and practice, are more centered around the first category of privacy and security and the fourth category of applicability. Future scholars, practitioners, and policy-makers have vast opportunities in the other much less exploited categories and the synthesis at the interface of multiple categories.\n\\section{The future}\n\\label{sec: the future}\nWhat are the gaps between the current design of blockchain and the design principle of a trust engine for a truly intelligent world? In this section, we do not intend to provide a comprehensive answer. Instead, we question the possibility of developing a plausible solution of singularity to the gaps without crossing the current boundaries of domain expertise by providing counterexamples.\n\n\\subsection*{Case Study 1: \\\\ How can we elaborate on the design principle of privacy and security for a better society?}\nThe scientific community of distributed systems and cryptography has made great milestones in blockchain development by achieving the anonymity of private information and the audibility of public transactions. However, privacy and security are the means but not necessarily the ends for a better society of human prosperity. Through history, critiques~\\cite{sep-privacy} have addressed privacy as the source of criminal activities, economic inefficiency, and the abuse of minorities. \\citet{cong2022crypto} identify that wash trading accounts for approximately 70\\% of the total cryptocurrency trading volumes. \\citet{cong2022anatomy} further provide a taxonomy of crimes enabled by the anonymity feature on blockchain. Abundant literature in behavioral science~\\cite{dawson2018study} establishes a connection between anonymity and abusive behavior. However, none of those negative effects on human behavior are considered in the original design principles of blockchain. How can we redesign privacy and security to cultivate human cooperation better and minimize abusive behavior? Behavior scientists have the expertise to contribute to this discussion. \n\n\\subsection*{Case Study 2: \\\\ How can we elaborate on the design principle of decentralization for a better society?}\nMost top-ranked blockchain projects aim to create a permissionless system that is more inclusive, democratic, and decentralized than those that have been presented previously. However, what is required to satisfy the permission for the permissionless blockchains? You must have access to the internet, afford the computer system requirements to run a replica node, and master the engineering skills needed to participate in the network. Unfortunately, according to World in Data, \\url{https:\/\/ourworldindata.org\/internet}, in most parts of our world, less than half of the population has access to the internet, let alone the higher requirements of computer system and engineering skills. Moreover, \\citet{ao2022decentralized} and the references therein evidence most of the crypto transactions are executed via centralized exchanges but not in the decentralized form of on-chain peer-to-peer transactions. \\citet{cong2022inclusion} further address the inclusion issues of the current Web3 economy due to technology barriers and lack of inclusion consideration in the original design principle of blockchain. The famous Irish playwright Bernard Shaw said, \"Revolutions have never lightened the burden of tyranny. They have only shifted it to another shoulder\"~\\cite{tiliouine2016state}. To direct blockchain development to avoid the fate stated by Bernard Shaw and serve a more decentralized society---of the people, by the people, and for the people---anthropologists, social scientists, and economists are able to contribute to designing a society of democracy, diversity, and inclusion. \n\n\\subsection*{Case Study 3: \\\\ How can we elaborate on the design principle of efficiency for a better society?}\n\nThe current solutions for efficiency or scalability on blockchain focus on improving transaction throughput and automating business processes. However, the pursuit of efficiency in an isolated system of short-sighted consideration might lead to inefficiency for the society as a whole with regard to long-term sustainability. For example, the current scalability solutions in general consider the blockchain system only in isolation but ignore the negative externality of the high-energy consumption design to the outside world~\\cite{truby2018decarbonizing}. Moreover, \\citet{grimmelmann2019all} notes that although smart contracts automate business processes, the current design ignores the ambiguity in the semantics of the programming language. Thus, a temporary convenience of the smart contract operation might lead to costly disputes in unforeseen scenarios when developers and users disagree. Policy-makers and lawyers can contribute to reconsidering the blockchain design to improve the efficiency of negotiations among different stakeholders and our society as a whole. \n\n\n\\subsection*{A Call for Collaboration}\n\\citet{kranzberg1986technology} stated the following in his 1985 address as president of the Society for the History of Technology (SHOT): \"Technology is neither good nor bad, nor is it neutral.\" blockchain is a powerful technology that brings about many new possibilities. How can we redesign blockchain for a truly intelligent world? To address this questions, we call for a joint endeavor from all disciplines. \n\n\\bibliographystyle{ACM-Reference-Format}\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}