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+{"text":"\\section{Introduction}\n\nA standard model (SM) like Higgs boson \\cite{higgs} (denoted as H(125) in this paper) was discovered at the Large Hadron Collider (LHC) in 2012.\nIn order to test the SM and discover possible physics beyond the SM (BSM),\nit is crucial to measure the Higgs Yukawa couplings and Higgs self couplings at the LHC and future high energy colliders.\nIn the first run of the LHC, the CMS and ATLAS has constrained $h\\bar t t$ Yukawa coupling indirectly through the global fit,\nwith a precision of  20$\\%$ and  30$\\%$  respectively \\cite{Khachatryan:2014jba,a}.\nWith 300\/fb, Yukawa couplings will be measured up to 23$\\%$, 13$\\%$  and 14$\\%$ for $h\\bar b b$, $h\\tau^+ \\tau^-$ and $h\\bar t t$\nrespectively \\cite{Peskin:2012we}.\nIt was also proposed to measure the top Yukawa coupling via the associated Higgs boson production with a single top quark\n\\cite{Barger:2009ky,Ellis:2013yxa,Biswas:2012bd,Biswas:2013xva,Farina:2012xp,Englert:2014pja,Chang:2014rfa}.\n The Higgs self coupling can be measured up to $50\\%$ at the LHC with 300\/fb \\cite{Peskin:2012we}.\nThere are extensive studies on measuring anomalous  triple Higgs coupling directly at the LHC \\cite{Baur:2002rb,Baur:2002qd,Baur:2003gp,Dolan:2012rv,Baglio:2012np} and\nfuture electron-positron colliders  \\cite{Baer:2013cma,Asner:2013psa}.\n\n\n\nFor the future high-luminosity electron-positron colliders, it is proposed to measure\nthe Higgs self coupling up to $28\\%$ for $\\sqrt{s_{e^+e^-}}=$ 240 GeV under the model-dependent assumption that only the Higgs self coupling is modified \\cite{McCullough:2013rea} .\nThe  precision of Higgs self coupling  can only be reached based on the precisely measured cross section of ZH associated production  up to $0.4\\%$ \\cite{Gomez-Ceballos:2013zzn}.\n Entering $e^+e^- \\rightarrow ZH$ via loops, the  triple Higgs coupling will be possibly polluted heavily by other anomalous couplings,\n and among them the dominant one is the $h-Z-Z$ coupling which appears even at tree-level.\n The first run results of LHC shows that the HVV couplings including $h-Z-Z$ coupling are consistent with those in the SM \\cite{Khachatryan:2014jba,a}.\n The Higgs-top coupling contributes to the process\n$e^+e^- \\rightarrow ZH$ via loops and is potentially important for triple Higgs coupling extraction.\nActually the full one-loop correction to $e^+e^- \\rightarrow ZH$ in the SM was calculated about two decades ago \\cite{Kniehl:1991hk,Denner:1992bc,Fleischer:1982af,Denner:1991ue}.\nIn this paper we will focus on the anomalous Higgs-top coupling, especially its effects on the extraction of  triple Higgs coupling.\n\nThis paper is arranged as following. In section II, we estimate the deviation of the cross section for the process $e^+e^- \\rightarrow ZH$ arising from anomalous Higgs-top coupling,\n and compare to that from triple Higgs coupling. In section III, we explore how to measure CP-violated Higgs-top coupling via the forward-backward asymmetry $A_{FB}$ for\nthe process $e^+e^- \\rightarrow ZH$. The last section contains our conclusion and discussion.\n\n\n\n\\section{ Pollution from Higgs-Top anomalous Coupling}\n\nIn the SM, the process $e^+e^- \\rightarrow ZH$ occurs at tree level and the Feynman diagram is shown in Fig. \\ref{fig1}.\n\\begin{figure}[!htbp]\n\\centering\n\\includegraphics[width=2.3in,totalheight=1.2in]{eehz.pdf}\n\\caption{Feyman diagram at tree-level for the process $e^+ e^- \\rightarrow Zh$}\n\\label{fig1}\n\\end{figure}\n\n\\begin{figure}[!htbp]\n\\includegraphics[width=2.3in,totalheight=1.2in]{eehz3h2.pdf}\n\\includegraphics[width=2.3in,totalheight=1.2in]{eehz3h.pdf}\n\\caption{Feynman diagram containing the anomalous $3h$ coupling, depicted as the black dot, at one-loop level for the process $e^+ e^- \\rightarrow Zh$. }\n\\label{fig2}\n\\end{figure}\nIn order to measure the triple Higgs coupling, one way is to produce the Higgs pair, provided that the center of mass energy of $e^+e^-$ is high enough via $ e^+e^- \\rightarrow HHZ$ or\n$ e^+e^- \\rightarrow HH \\nu \\bar \\nu$ \\cite{Levy:2015fva}. For such processes, the cross sections are notorious small.\nHigh energy and high luminosity are both required. Another way to measure the\ntriple Higgs coupling is via the virtual effects which are shown in Fig. \\ref{fig2}.\nThe capacity of measuring triple Higgs has been estimated by ref \\cite{McCullough:2013rea}. For completeness we recalculate the analytical result for\n$$\\delta_{\\sigma}\\equiv \\frac{\\Delta \\sigma}{\\sigma}=\\frac{\\sigma_{\\delta_h \\ne 0}-\\sigma_{\\delta_h = 0}}{\\sigma_{\\delta_h = 0}}$$ from the triple Higgs coupling\n$C_{SM}(1+ \\delta_h) H H H=-3i \\frac{m^2_h}{v}(1+ \\delta_h) H H H $ as\n\\begin{equation}\n\\begin{split}\n  \\delta_{\\sigma}(3h) =\\frac{3 \\alpha m_h^2 \\delta_h} {16 \\pi \\beta c_w^2 s_w^2 m_z^2}\n                    Re\\Big[&2\\rho \\Big (C_1(m_h^2)+C_{11}(m_h^2)+C_{12}(m_h^2) \\Big)\\\\\n                    & -\\beta \\Big (B_0-4 C_{00}(m_h^2)+4 m_z^2C_0(m_h^2)+3m_h^2 B'_0 \\Big ) \\Big],\n\\end{split}\n\\end{equation}\nwhere $\\delta_h=0$ corresponds to the case in the SM.\nHere $$ \\beta=m_h^4-2 m_h^2 (m_z^2+s)+m_z^4+10 m_z^2 s+s^2,$$\n$$ \\rho=(m_h^2-m_z^2-s)\\left((m_h-m_z)^2-s\\right)\\left((m_h+m_z)^2-s\\right)$$\nThe definition of the one loop scalar functions B, C etc. can be found in Ref. \\cite{Ellis:2007qk} and\n$ C_0(m_h^2)=C_0(m_h^2,m_z^2,s,m_h^2,m_h^2,m_z^2)$,\n$ C(m_h^2)=C(m_h^2,s,m_z^2,m_h^2,m_h^2,m_z^2)$,\n$ B_0=B_0(m_h^2,m_h^2,m_h^2)$,\n$ B'_0=\\frac{\\partial B_0}{\\partial p^2}\\bigg|_{p^2=m_h^2}$.\n\n\\begin{figure}[!htbp]\n\\includegraphics[width=2.3in,totalheight=1.2in]{eehzhtt1.pdf}\n\\includegraphics[width=2.3in,totalheight=1.2in]{eehzhtt2.pdf}\n\\caption{Feynman diagram containing the anomalous $h\\bar t t$ coupling, depicted as the black dot, at one-loop level for the process $e^+ e^- \\rightarrow Zh$.}\n\\label{fig3}\n\\end{figure}\nIn this paper we will calculate the contributions from Higgs-top coupling which are shown in Fig. \\ref{fig3}\n\\footnote{In fact,the contributions from $Z\/\\gamma-H$ bubble transition diagrams are zero.}. The Higgs-top coupling can be parameterized as\n$$\nC_{SM}(1+ \\delta_t) H \\bar t t=-i \\frac{m_t}{v}(1+ \\delta_t) H \\bar t t,\n$$\nwhere $\\delta_t=0$ corresponds to the case in the SM.\n\nThe analytical results can be written as\n\\begin{equation}\n \\begin{split}\n &\\delta_{\\sigma}(htt)\\\\\n  =&-\\frac{4 N_c\\alpha m_t^2 (s-m_z^2) v_1 v_2 \\delta_t}{3 \\pi s \\beta m_z^2 (v_1^2+a_1^2)}Re\\Big[\\beta \\Big (B_0(m_h^2)-4 C_{00}(m_t^2) \\Big)\\\\\n  &-2 \\rho \\Big(C_1(m_t^2)+C_{11}(m_t^2)+C_{12}(m_t^2) \\Big)-6 m_z^2 s \\big(s+m_z^2-m_h^2 \\big ) C_0(m_t^2)\\Big] \\\\\n                       &+\\frac{N_c \\alpha m_t^2 \\delta_t}{\\pi c_w^2 s_w^2 m_z^2 \\beta}  Re\\Big[ \\beta \\Big( 2 \\big (v_2^2+a_2^2 \\big )\\big(B_0(m_h^2)- 4 C_{00}(m_t^2)\\big) +2 a_2^2 \\big (B_0(s)+B_0(m_z^2) \\big) \\Big)  \\\\\n                        &-\\rho\\Big( 4 \\big (v_2^2+a_2^2\\big )\\big (C_1(m_t^2)+C_{11}(m_t^2)+C_{12}(m_t^2)\\big)+2a_2^2 C_2(m_t^2)  \\Big)\\\\\n                        &+\\Big(\\big (v_2^2+a_2^2 \\big) \\big((4 m_t^2- m_z^2-s) \\beta - \\rho \\big) +\\big (v_2^2-a_2^2 \\big) \\big (m_h^2-4 m_t^2 \\big ) \\beta \\Big) C_0(m_t^2)  \\Big]\\\\\n                      &+\\frac{N_c\\alpha m_t^2 \\delta_t}{4 \\pi c_w^2 s_w^2 m_z^2}Re\\Big[-B_0(m_h^2)+\\big(4 m_t^2- m_h^2 \\big) B'_0(m_h^2)\\Big]\n \\end{split}\n \\label{eq2}\n\\end{equation}\nIn Eq. (\\ref{eq2}), the first\/second\/third terms are from the contributions of the diagram with photon propagator\/Z boson propagator\/the counter term of ZZH vertex, respectively.\nHere\n$ \\alpha = \\frac{e^2}{4\\pi}$,\n$N_c=3$,\n$ v_1=-\\frac{1}{4}+s_w^2$,\n$ a_1=\\frac{1}{4}$,\n$ v_2=\\frac{1}{4}-\\frac{2}{3}s_w^2$,\n$ a_2=-\\frac{1}{4}$,\n$ C_0(m_t^2)=C_0(m_h^2,m_z^2,s,m_t^2,m_t^2,m_t^2)$,\n$ C(m_t^2)=C(m_h^2,s,m_z^2,m_t^2,m_t^2,m_t^2)$,\n$ B_0(m_h^2)=B_0(m_h^2,m_t^2,m_t^2)$,\n$ B_0(m_z^2)=B_0(m_z^2,m_t^2,m_t^2)$,\n$ B_0(s)=B_0(s,m_t^2,m_t^2)$.\n\n\n\nWe use LoopTools \\cite{Hahn:1998yk} to do the scalar integral for different  c.m. energies.\nIn Fig. \\ref{fig4}, we show the deviation of cross section arising from $\\delta_t$ and  $\\delta_h$ as a function of $\\sqrt{s}_{e^+ e^-}$.\nSeveral numerical results for the typical c.m. energy are\n\\begin{equation}\n \\delta_\\sigma^{240,350,400,500}=1.45,0.27,0.05,-0.19\\times \\delta_h \\%\n\\end{equation}\n \\begin{equation}\n \\delta_\\sigma^{240,350,400,500}=-0.49,1.38,2.14,2.12\\times \\delta_{t} \\%\n\\end{equation}\n\\begin{figure}[!htbp]\n\n\\includegraphics[width=0.3\\textwidth]{htthhh.pdf}\n\\caption{ Relative correction $\\delta_\\sigma$  due to anomalous $h\\bar{t}t$-coupling $\\delta_t$ (red)\n         and anomalous triple Higgs coupling $\\delta_h$ (blue), as a function of the $e^+ e^-$ center-of-mass (c.m.) energy from 220 GeV to 500 GeV. Note that the precision of\n         relative correction can reach $0.4\\%$ for high luminosity $e^+ e^-$ colliders. }\n\\label{fig4}\n\\end{figure}\n\nThe figures show that the behavior for $\\delta_t$ and $\\delta_h$ is opposite. At low energy end, the relative correction $\\delta_\\sigma$ happen to be dominant by $\\delta_h$, on\nthe contrary for the high energy end, the $\\delta_\\sigma$ arising from anomalous Higgs-top coupling can't be neglected.\nFor the proposed collider of Circular Electron-Positron Collider with $\\sqrt{s}_{e^+e^-} \\simeq 240$ GeV, the extraction\nof triple Higgs coupling is  polluted by Higgs-top coupling. For the International Linear Collider with option of high energy,\nthe pollution from Higgs-top coupling must be taken into account.\n\n\\section{ Measuring CP-violated Higgs-Top Coupling}\n\nThough the newly discovered Higgs boson H(125) is SM-like, it does not exclude the possibility that H(125) is CP mixing state. As emphasized by \\cite{zhu,Mao:2014oya} that\nCP spontaneously broken \\cite{Lee} may be closely related to the lightness of the H(125). In fact, current measurements are insensitive to the mixing, especially for\nH decaying into gauge bosons since the CP violation usually entering the couplings via loops.\n\nIn this paper we parameterize the CP violation through\n$$\nC_{SM} H\\left(1+\\delta_t +i \\delta_a \\gamma_5\\right)=-i \\frac{m_t}{v} H\\left(1+\\delta_t +i \\delta_a \\gamma_5\\right).\n$$\nIndirect constraints on $\\delta_t$ and $\\delta_a$ at the LHC have been studied in \\cite{Ellis:2013yxa}.\nAt the 68\\% CL  the allowed region for ($1+\\delta_t$ , $\\delta_a$) is a crescent with apex close to the SM point(1,0) \\cite{Ellis:2013yxa}.\nThe parameter space close to the SM point, namely $\\delta_t \\rightarrow 0$ and $\\delta_a \\rightarrow 0$ is allowed. At the same time, the parameter space with\nboth non-zero $\\delta_t,  \\delta_a$ is also allowed. In fact, it is quite challenging for LHC to completely exclude the latter case via the indirect method.\nOn the contrary, based on the last section analysis, the cross section deviation depends only on $\\delta_t$ but not $\\delta_a$. This point will be made clear below.  Therefore\nit is important to explore the method to measure the $\\delta_a$ at electron-positron collider.\n\nThe analytical results for the differential cross section arising from $\\delta_a$ can be written as\n\\begin{equation}\n \\begin{split}\n &\\frac{1}{\\delta_a} \\frac{d\\sigma}{d cos\\alpha}\\\\\n =&\\frac{32 N_c a_1 m_t^2 \\pi \\alpha^3 cos\\alpha \\sqrt{\\left((m_h-m_z)^2-s\\right) \\left((m_h+m_z)^2-s\\right)}}{c^4_w s^4_w \\left(m_z^2-s\\right)}\\\\\n &Im\\Big[\\frac{1}{3} v_2  C_0(m_t^2)\n   +\\frac{ s}{c_w^2 s_w^2 \\left(m_z^2-s\\right)}  v_1\n\\big((v_2^2+a_2^2) C_0(m_t^2)+2 a_2^2 C_2(m_t^2)\\big)\\Big]\n \\end{split}\n\\end{equation}\nHere $cos\\alpha$ is the angle between the momentum of the electron and the Z boson. The differential cross section is  proportional to $cos\\alpha$, which is\ndue to the term $\\varepsilon_{\\mu \\nu \\rho \\lambda}  \\varepsilon^{\\mu \\nu \\alpha \\beta}  p_2^{\\rho}  p_1^{\\lambda}  k_{1\\alpha}  k_{2\\beta}$\nwhere $p_1$ $p_2$ are the momentum of electron and positron and $k_1$ $k_2$ are the momentum of Higgs and Z. Another critical requirement for non-vanishing contribution\nto the differential cross section\nis that there should be imaginary part from top loops. This requires that the $\\sqrt{s}_{e^+e^-}$ must be great than $2 m_t$.\n\n\nIt is obvious that the CP-odd contributions to the total cross section is zero.\nIn order to show the different contributions from $\\delta_t$ and $\\delta_a$ respectively, we plot the normalized differential cross sections for several $\\sqrt{s}_{e^+e^-}$\nand set the corresponding parameter $\\delta_t$ or $\\delta_a$ equal to 1.\n\\begin{figure}[!htbp]\n\\includegraphics[width=0.5\\textwidth]{total4.pdf}\n\\caption{Differential scattering cross section as a function of the scattering angle with $\\sqrt{s}=240GeV$(orange),$350GeV$(red),$400GeV$(green),$500GeV$(blue).\nAnd solid\/dashed lines stand for the contributions\n from  $\\delta_t$\/$\\delta_a$ respectively. }\n\\label{6}\n\\end{figure}\nFrom the  figure, it is quite clear that the differential cross sections arising from $\\delta_t$ are symmetric and anti-symmetric from $\\delta_a$.\nFor $\\sqrt{s}=240GeV$, the contribution from $\\delta_a$ is zero because there is no imaginary part of $ C_0(m_t^2)$.\nWhen $\\sqrt{s}_{e^+e^-} > 2 m_t$ there are nonzero contributions from $\\delta_a$ as expected.\n\nIn order to gauge the forward-backward asymmetry, we introduce\n$$\nA_{FB} \\equiv \\frac{\\int_{0}^1 d\\cos \\alpha \\frac{d\\sigma}{d\\cos\\alpha}  -  \\int_{-1}^0 d\\cos \\alpha \\frac{d\\sigma}{d\\cos\\alpha} }{\\sigma_{tot} }\n$$\n\nIn Fig.  \\ref{fig9}, we plot $A_{FB}$ as a function of $\\sqrt{s}_{e^+e^-}$ with $\\delta_a=1$ for polarized and unpolarized electron\/positron beam.\n\n\\begin{figure}[!htbp]\n\\includegraphics[width=0.4\\textwidth]{afbsum.pdf}\n\\caption{  $A_{FB}$ as a function of $\\sqrt{s}_{e^+e^-}$ from $220GeV$ to $500GeV$ for polarized or unpolarized electron\/positron beams.\nThe black line, blue dashed and red dashed seperately\ncorrespond to  unpolarized electron\/positron beams,$e_R^+ e_L^-$ and $e_L^+ e_R^-$ polarizations.}\n\\label{fig9}\n\\end{figure}\n\n\nFrom the figure we can see that the asymmetry can reach $0.7 \\%$ for $\\sqrt{s}_{e^+e^-}$. Such precision is comparable to that of cross section measurement.\nIt seems that the high luminosity collider is necessary.\n\n\n\n\\section{Conclusion and discussion}\n\nIn this paper, we explore the Higgs-top anomalous coupling pollution to the extraction of Higgs self coupling via precisely measuring cross section of $e^+e^- \\rightarrow ZH$.\nThe important conclusion is that the pollution is small for the $\\sqrt{s}_{e^+e^-} =240$ GeV, but can be sizable for higher energy collider.\nThe contributions to total cross section from Higgs-top CP-odd coupling is vanishing, while such interaction can\n be scrutinized via forward-backward asymmetry for $\\sqrt{s}_{e^+e^-}$ greater than $2 m_t$.\n\n\n\\section*{Acknowledgement}\n\n We would like to thank Shao Long Chen , Gang Li and Pengfei Yin for the useful discussions. This work was supported in part by the Natural Science Foundation\n of China (Nos. 11135003 and 11375014).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{E:introduction}\n\nWormholes are handles or tunnels in spacetime\nconnecting widely separated regions of our\nUniverse or different universes altogether.\nWhile there had been some forerunners,\nmacroscopic traversable wormholes were first\ndiscussed in detail by Morris and Thorne\n\\cite{MT88} in 1988.  A few years later,\nSung-Won Kim \\cite{Kim} proposed the\npossible existence of an evolving wormhole\nin the context of the\nFriedmann-Lemaitre-Robertson-Walker (FLRW)\ncosmological model by assuming that the\nmatter content can be divided into two\nparts, the cosmological part that depends on\ntime only and the wormhole part that depends\non space only.  The discussion was later\nexpanded by Cataldo et al. \\cite{mC13}.\n\nThe purpose of this paper is to study the\nrelationship between wormholes inspired by\nnoncommutative geometry and the Kim model.\nThe noncommutative-geometry background\nsimultaneously affects both the wormhole\nconstruction and the cosmological part of\nthe solution.  This result differs\nsignificantly from the outcomes in Refs.\n\\cite{Kim} and \\cite{mC13}.\n\nRegarding the strategy, Ref. \\cite{MT88}\nconcentrates mainly on the wormhole\ngeometry by specifying the metric\ncoefficients.  This strategy requires a\nsearch for matter or fields that can produce\nthe energy-momentum tensor needed to sustain\nthe wormhole.  Here it needs to be emphasized\nthat we are able to satisfy the geometric\nrequirements from the physical properties.\nThe result is an evolving zero-tidal force\nwormhole solution; it is restricted to the\ncurvature parameters $k=0$ and $k=-1$,\ncorresponding to an open Universe.\n\nViewed from a broader perspective, it has\nalready been shown that noncommutative\ngeometry, which is an offshoot of string\ntheory, can account for the flat galactic\nrotation curves \\cite{pK17, fR12}, but\nunder certain conditions, noncommutative\ngeometry can also support traversable\nwormholes \\cite{KG14, pK18, pK16, pK15,\n Jamil14, FKRI12}.\n\nThis paper is organized as follows: Sec.\n\\ref{S:structure} briefly recalls the\nstructure of wormholes and the basic\nfeatures of noncommutative geometry.\nSec. \\ref{S:Kim} continues with the\nSung-Won Kim model. Here the discussion\nis necessarily more detailed, partly\nin the interest of completeness, but mainly\nto allow the inclusion of a more general\nform of the Einstein field equations.\nThese are subsequently used in Sec. \\ref\n{S:special} to obtain a wormhole solution\nthat does not depend on the separation\nof the matter content.  In Sec. \\ref{S:nc}\nwe derive a wormhole solution from the\nnoncommutative-geometry background,\nfollowed by a discussion of the null\nenergy condition in Sec. \\ref {S:violation}.\nSec. \\ref{S:comparison} features a\n comparison to an earlier solution.\n In Sec. \\ref{S:conclusion}, we conclude.\n\n\n\\section{Wormhole structure and\n   noncommutative geometry}\\label{S:structure}\n\nMorris and Thorne \\cite{MT88} proposed the\nfollowing static and spherically symmetric\nline element for a wormhole spacetime:\n\\begin{equation}\\label{E:line1}\nds^{2}=-e^{2\\Phi(r)}dt^{2}+\\frac{dr^2}{1-b(r)\/r}\n+r^{2}(d\\theta^{2}+\\text{sin}^{2}\\theta\\,\nd\\phi^{2}),\n\\end{equation}\nusing units in which $c=G=1$.  Here $b=b(r)$\nis called the \\emph{shape function} and\n$\\Phi=\\Phi(r)$ is called the \\emph{redshift\nfunction}, which must be everywhere finite\nto avoid an event horizon.  For the shape\nfunction we must have $b(r_0)=r_0$, where\n$r=r_0$ is the radius of the \\emph{throat}\nof the wormhole.  The wormhole spacetime\nshould be asymptotically flat, i.e.,\n$\\text{lim}_{r\\rightarrow \\infty}\\Phi(r)\n=0$ and  $\\text{lim}_{r\\rightarrow \\infty}\nb(r)\/r=0$.  An important requirement is the\n\\emph{flare-out condition} at the throat:\n$b'(r_0)<1$, while $b(r)<r$ near the\nthroat.  The flare-out condition can only\nbe met by violating the null energy\ncondition (NEC), which states that\n\\begin{equation}\n  T_{\\alpha\\beta}k^{\\alpha}k^{\\beta}\\ge 0\n\\end{equation}\nfor all null vectors $k^{\\alpha}$, where\n$T_{\\alpha\\beta}$ is the energy-momentum\ntensor.  Matter that violates the NEC is\ncalled ``exotic\" in Ref. \\cite{MT88}.  In\nparticular, for the outgoing null vector\n$(1,1,0,0)$, the violation has the form\n\\begin{equation}\n   T_{\\alpha\\beta}k^{\\alpha}k^{\\beta}=\n   \\rho +P^r<0.\n\\end{equation}\nHere $T^t_{\\phantom{tt}t}=-\\rho$ is the energy\ndensity, $T^r_{\\phantom{rr}r}= P^r$ is the\nradial pressure, and\n$T^\\theta_{\\phantom{\\theta\\theta}\\theta}=\nT^\\phi_{\\phantom{\\phi\\phi}\\phi}=P^t$ is\nthe lateral pressure.\n\nReturning now to the noncommutative-geometry\nbackground mentioned earlier, we need to\nrecall that, as an offshoot of string theory,\nnoncommutative geometry replaces point-like\nparticles by smeared objects.  (For a detailed\ndiscussion, see Refs. \\cite {SS03, NSS06,\nNS10}.)  As a result, spacetime can be\nencoded in the commutator\n$[\\textbf{x}^{\\mu},\\textbf{x}^{\\nu}]\n=i\\theta^{\\mu\\nu}$, where $\\theta^{\\mu\\nu}$ is\nan antisymmetric matrix that determines the\nfundamental cell discretization of spacetime\nin the same way that Planck's constant\ndiscretizes phase space \\cite{NSS06}.  An\ninteresting and effective way to model the\nsmearing effect, discussed in Refs.\n\\cite{pK15,LL12, NM08}, is to assume that\nthe energy density of the static,\nspherically symmetric, smeared, and\nparticle-like gravitational source is\ngiven by\n\\begin{equation}\\label{E:rho}\n  \\rho(r)=\\frac{\\mu\\sqrt{\\beta}}\n     {\\pi^2(r^2+\\beta)^2},\n\\end{equation}\nwhich can be interpreted to mean that the\ngravitational source causes the mass $\\mu$\nof a particle to be diffused throughout the\nregion of linear dimension $\\sqrt{\\beta}$\ndue to the uncertainty; so $\\sqrt{\\beta}$\nhas units of length.\n(Ref. \\cite{NSS06} uses a Gaussian\ndistribution instead of Eq. (\\ref{E:rho})\nto represent $\\rho$.)  Eq. (\\ref{E:rho})\nleads to the mass distribution\n\\begin{equation}\\label{E:source}\n   \\int^r_04\\pi(r')^2\\rho(r')dr'=\n   \\frac{2M\\sqrt{\\beta}}{\\pi}\n   \\left(\\frac{1}{\\sqrt{\\beta}}\\text{tan}^{-1}\n   \\frac{r}{\\sqrt{\\beta}}-\n   \\frac{r}{r^2+\\beta}\\right),\n\\end{equation}\nwhere $M$ is now the total mass of the source.\n\nAccording to Ref. \\cite{NSS06}, noncommutative\ngeometry is one of the basic properties of\nspacetime that does not depend on particular\nfeatures such as curvature.  Moreover, since\nthe noncommutative effects can be implemented\nby modifying only the energy-momentum tensor,\nthere is no need to change the Einstein tensor\nin the field equations.  As a result, the\nlength scales need not be microscopic.\n\n\\section{The Sung-Won Kim model}\\label{S:Kim}\n\nThe Sung-Won Kim cosmological model with a\ntraversable wormhole is given by \\cite{Kim}\n\\begin{equation}\\label{E:line2}\nds^{2}=-e^{2\\Phi(r)}dt^{2}+[R(t)]^2\\left[\n\\frac{dr^2}{1-kr^2-b(r)\/r}\n+r^{2}(d\\theta^{2}+\\text{sin}^{2}\\theta\\,\nd\\phi^{2})\\right],\n\\end{equation}\nwhere $R(t)$ is the scale factor of the\nUniverse and $k$ is the sign of the\ncurvature of spacetime, i.e., $k=+1$,\n0, or $-1$.   (So while $b=b(r)$ is still\nthe shape function, it is subject to\nconditions that are different from those\nof a Morris-Thorne wormhole.)  The\nEinstein field equations in Ref. \\cite\n{Kim} are based on Eq. (\\ref{E:line2}),\nbut it is subsequently assumed that\n$\\Phi(r)\\equiv 0$ to be consistent with\nthe FLRW model.\n\nThe discussion of the Sung-Won Kim model\nis continued and elaborated on in Ref.\n\\cite{mC13}.  Unfortunately, the field\nequations in Refs. \\cite{Kim} and\n\\cite{mC13} do not agree and need to be\nrederived.  Here we follow Ref. \\cite{Kim}\nand base the calculations on line element\n(\\ref{E:line2}).  The results are\n\\begin{equation}\\label{E:Einstein1}\n   8\\pi\\rho(r,t)=3\\left(\\frac{\\dot{R}}{R}\\right)^2\n   e^{-2\\Phi(r)}+\\frac{3k}{R^2}+\\frac{b'(r)}{R^2r^2},\n\\end{equation}\n\\begin{equation}\\label{E:Einstein2}\n   8\\pi P^r(r,t)=-2\\frac{\\ddot{R}}{R}e^{-2\\Phi(r)}\n   -\\left(\\frac{\\dot{R}}{R}\\right)^2e^{-2\\Phi(r)}\n   -\\frac{k}{R^2}-\\frac{b(r)}{R^2r^3}+\\frac{2}{R^2r}\n   \\Phi'(r)\\left(1-kr^2-\\frac{b(r)}{r}\\right),\n\\end{equation}\n\\begin{multline}\\label{E:Einstein3}\n   8\\pi P^t(r,t)=-2\\frac{\\ddot{R}}{R}e^{-2\\Phi(r)}\n   -\\left(\\frac{\\dot{R}}{R}\\right)^2e^{-2\\Phi(r)}\n   -\\frac{k}{R^2}+\\frac{b(r)-rb'(r)}{2R^2r^3}\\\\\n   +\\frac{1}{R^2}\\left[(\\Phi'(r))^2\\left(1-kr^2-\n   \\frac{b(r)}{r}\\right)+\\Phi''(r)\\left(1-kr^2-\n   \\frac{b(r)}{r}\\right)\\right.\\\\\n   \\left.-\\frac{1}{2}\\Phi'(r)\\left(2kr\n   +\\frac{rb'(r)-b(r)}{r^2}\\right)\\right]\n   +\\frac{1}{R^2r}\\Phi'(r)\\left(1-kr^2\n   -\\frac{b(r)}{r}\\right),\n\\end{multline}\n\\begin{equation}\\label{E:Einstein4}\n  8\\pi T_{01}=\\frac{\\dot{R}}{R^2}\n  e^{-\\Phi(r)}\\Phi'(r)\n  \\left(1-kr^2-\\frac{b(r)}{r}\\right)^{1\/2},\n\\end{equation}\nwhere $T_{01}$ is the outward energy flow.\n(The prime and overdots denote the derivatives\nwith respect to $r$ and $t$, respectively.)\n\nIf $\\Phi(r)\\equiv 0$, the results agree with\nthose in Ref. \\cite{mC13} (omitting $\\Lambda$,\nthe cosmological constant).  For convenience,\nthese will now be restated:\n\\begin{equation}\\label{E:E1}\n   8\\pi\\rho(r,t)=3\\left(\\frac{\\dot{R}}{R}\\right)^2\n   +\\frac{3k}{R^2}+\\frac{b'}{R^2r^2},\n\\end{equation}\n\\begin{equation}\\label{E:E2}\n   8\\pi P^r(r,t)=-2\\frac{\\ddot{R}}{R}-\n   \\left(\\frac{\\dot{R}}{R}\\right)^2\n   -\\frac{k}{R^2}-\\frac{b}{R^2r^3},\n\\end{equation}\n\\begin{equation}\\label{E:E3}\n   8\\pi P^t(r,t)=-2\\frac{\\ddot{R}}{R}\n   -\\left(\\frac{\\dot{R}}{R}\\right)^2\n   -\\frac{k}{R^2}+\\frac{b-rb'}\n   {2R^2r^3}.\n\\end{equation}\n\nThe next step depends on a key insight,\ndue to Sung-Won Kim \\cite{Kim}, that\nallows the separation of the Einstein\nfield equations into two parts, namely,\nthe following ansatz for the matter\nparts:\n\\begin{equation}\\label{E:Kim1}\n   R^2(t)\\rho(r,t)=R^2(t)\\rho_c(t)\n       +\\rho_w(r),\n\\end{equation}\n\\begin{equation}\\label{E:Kim2}\n   R^2(t)P^r(r,t)=R^2(t)P_c(t)\n      +P^r_w(r),\n\\end{equation}\n\\begin{equation}\\label{E:Kim3}\n   R^2(t)P^t(r,t)=R^2(t)P_c(t)\n      +P^t_w(r).\n\\end{equation}\nThe subscripts $c$ and $w$ refer,\nrespectively, to the cosmological\nand wormhole parts.  So $P_c$\nnecessarily represents the isotropic\npressure.\n\nThe ansatz now allows us to separate\nEqs. (\\ref{E:E1})-(\\ref{E:E3}) into\ntwo parts, the left side being a\nfunction of $t$ and the right side\na function of $r$, also carried out in\nRef. \\cite{mC13}:\n\\begin{equation}\\label{E:separate1}\n   R^2\\left[8\\pi\\rho_c-3\\left(\\frac{\\dot{R}}\n   {R}\\right)^2-\\frac{3k}{R^2}\\right]=\n   \\frac{b'}{r^2}-8\\pi\\rho_w=l,\n\\end{equation}\n\\begin{equation}\\label{E:separate2}\n  R^2\\left[8\\pi P_c+2\\frac{\\ddot{R}}{R}\n  +\\left(\\frac{\\dot{R}}{R}\\right)^2\n  +\\frac{k}{R^2}\\right]=-\\frac{b}{r^3}\n  -8\\pi P^r_w=m,\n\\end{equation}\n\\begin{equation}\\label{E:separate3}\n   R^2\\left[8\\pi P_c+2\\frac{\\ddot{R}}{R}\n   +\\left(\\frac{\\dot{R}}{R}\\right)^2\n   +\\frac{k}{R^2}\\right]=\n   \\frac{b-rb'}{2r^3}-8\\pi P^t_w=m,\n\\end{equation}\nwhere $l$ and $m$ are constants.  The\nreason is that a function of $t$ cannot\nbe equal to a function of $r$ for all\n$t$ and $r$ unless they are equal to\nsome constant.  In Eqs. (\\ref{E:separate2})\nand (\\ref{E:separate3}), the constants\nare the same since the cosmological parts\nare equal.  The constants $l$ and $m$ may\nbe taken as arbitrary.\n\nReturning now to Eqs.\n(\\ref{E:Einstein1})-(\\ref{E:Einstein4}),\nrecall that we obtained the more general\nform of the field equations by using\nline element (\\ref{E:line2}), as\nsuggested in Ref. \\cite{Kim}.  It now\nbecomes apparent, however, that the\nseparation in Eqs.\n(\\ref{E:separate1})-(\\ref{E:separate3})\ncannot be carried out by means of Eqs.\n(\\ref{E:Kim1})-(\\ref{E:Kim3}) unless\nwe assume that $\\Phi(r)\\equiv 0$, which\ntakes us back to the FLRW model.\n\n\\section{The noncommutative wormhole}\n   \\label{S:nc}\nTo obtain a wormhole solution, we will\nconsider the special case $l=-3m$,\nfollowing Ref. \\cite{mC13}.  Eqs.\n(\\ref{E:separate1})-(\\ref{E:separate3})\nthen yield\n\\begin{equation}\\label{E:R1}\n  3\\left(\\frac{\\dot{R}}{R}\\right)^2\n   +\\frac{3k}{R^2}-\\frac{3m}{R^2}\n   =8\\pi\\rho_c\n\\end{equation}\nand\n\\begin{equation}\\label{E:R2}\n   -2\\frac{\\ddot{R}}{R}-\n   \\left(\\frac{\\dot{R}}{R}\\right)^2\n   -\\frac{k}{R^2}+\\frac{m}{R^2}\n   =8\\pi P_c\n\\end{equation}\nfor the cosmological part, while the\nwormhole part is given by\n\\begin{equation}\\label{E:W1}\n   \\frac{b'}{r^2}-8\\pi\\rho_w=-3m,\n\\end{equation}\n\\begin{equation}\\label{E:W2}\n   -\n\\frac{b}{r^3}-8\\pi P^r_w=m,\n\\end{equation}\n\\begin{equation}\\label{E:W3}\n   \\frac{b-rb'}{2r^3}-8\\pi P^t_w=m.\n\\end{equation}\n\nThe wormhole solution can be obtained\nfrom Eqs. (\\ref{E:W1})-(\\ref{E:W3}) by\nmaking use of Eq. (\\ref{E:rho}).  It\nremains to be seen what restrictions\nwill be placed on the solutions due to\nthe cosmological part and to the\nnecessary violation of the NEC.  But\nfor now we have from Eq. (\\ref{E:W1})\nand Eq. (\\ref{E:rho}) that\n\\begin{equation}\n    b'(r)=8\\pi\\frac{\\mu\\sqrt{\\beta}r^2}\n    {\\pi^2(r^2+\\beta)^2}-3mr^2.\n\\end{equation}\nIntegrating and using the condition $b(r_0)\n=r_0$, we obtain\n\\begin{multline}\\label{E:shape2}\n  b(r)=\\frac{4M\\sqrt{\\beta}}{\\pi}\n  \\left(\\frac{1}{\\sqrt{\\beta}}\\text{tan}^{-1}\n  \\frac{r}{\\sqrt{\\beta}}-\\frac{r}{r^2+\\beta}\n  \\right)-mr^3\\\\\n  -\\frac{4M\\sqrt{\\beta}}{\\pi}\n  \\left(\\frac{1}{\\sqrt{\\beta}}\\text{tan}^{-1}\n  \\frac{r_0}{\\sqrt{\\beta}}-\\frac{r_0}{r_0^2\n  +\\beta}\\right)+mr_0^3+r_0,\n\\end{multline}\nwhere $M$ is the mass of the wormhole.\nIt now becomes apparent that the resulting\nspacetime is not asymptotically flat.  The\nnormal procedure is to cut off the wormhole\nmaterial at some $r=a$ and then join the\nstructure to an external Schwarzschild\nspacetime. We will see in the next section,\nhowever, that the need to violate the NEC\n($\\rho +P^r<0$) requires a slight\nmodification of the shape function,\nresulting in the required asymptotic\nflatness.\n\n\\section{Violating the NEC}\\label{S:violation}\nTo check the violation of the NEC\n($\\rho +P^r<0$) for the wormhole, we\nlet $R\\equiv \\text{constant}$\nand obtain from Eqs. (\\ref{E:E1}) and\n(\\ref{E:E2}),\n\\begin{equation}\n   b-rb'-2kr^3>0.\n\\end{equation}\nAt $r=r_0$, we therefore get\n\\begin{equation}\n   r_0-r_0\\frac{8\\pi\\mu\\sqrt{\\beta}r_0^2}\n   {\\pi^2(r_0^2+\\beta)^2}+3mr_0^3\n   -2kr_0^3>0.\n\\end{equation}\nSince $\\sqrt{\\beta}$ is extremely small, we\nactually have\n\\begin{equation}\\label{E:NEC}\n   r_0+(3m-2k)r_0^3\\gtrsim 0.\n\\end{equation}\nTo check this condition, we need to return\nto Ref. \\cite{NSS06} for some additional\nobservations.  The relationship between\nthe radial pressure and energy density\nis given by\n\\begin{equation}\\label{E:EoS}\n   P^r=-\\rho.\n\\end{equation}\nThe reason is that the source is a\nself-gravitating droplet of anisotropic\nfluid of density $\\rho$ and the radial\npressure is needed to prevent the\ncollapse back to the matter point.  In\naddition, the  lateral pressure is\ngiven by\n\\begin{equation}\\label{E:tr1}\n   P^t=-\\rho-\\frac{r}{2}\n     \\frac{\\partial\\rho}{\\partial r}.\n\\end{equation}\nSince the length scales can be\nmacroscopic, we can retain Eq.\n(\\ref{E:EoS}) and then use Eq.\n(\\ref{E:tr1}) to write\n\\begin{equation}\\label{E:tr2}\n    P^t=-\\rho-\\frac{r}{2}\n     \\frac{\\partial\\rho}{\\partial r}\n     =P^r+\\frac{2\\mu r^2\\sqrt{\\beta}}\n     {\\pi^2(r^2+\\beta)^3}\n\\end{equation}\nby Eq. (\\ref{E:rho}).  So on larger\nscales, we have $P^r=P^t$.  Since the\npressure becomes isotropic, we can\nassume the equation of state to be\n$P_c=-\\rho_c$.  Substituting in Eqs.\n(\\ref{E:R1}) and (\\ref{E:R2}), we get\n\\begin{equation*}\n   -2\\frac{\\ddot{R}}{R}\n   +2\\left(\\frac{\\dot{R}}{R}\\right)^2\n   +\\frac{2k}{R^2}-\\frac{2m}{R^2}=0.\n\\end{equation*}\nThis equation can be rewritten as\n\\begin{equation}\n   3\\frac{\\ddot{R}}{R}\n   -3\\left(\\frac{\\dot{R}}{R}\\right)^2=\n   \\frac{3k}{R^2}-\\frac{3m}{R^2}.\n\\end{equation}\nSubtracting the Friedmann equations\n\\[\n   3\\frac{\\ddot{R}}{R}=-4\\pi(\\rho_c\n      +3P_c)\n\\]\nand\n\\[\n   3\\left(\\frac{\\dot{R}}{R}\\right)^2\n   =8\\pi\\rho_c-\\frac{3k}{R^2}\n\\]\nnow yields\n\\begin{equation}\n   \\frac{3k}{R^2}-\\frac{3m}{R^2}=\n   -4\\pi(\\rho_c+3P_c)-8\\pi\\rho_c\n   +\\frac{3k}{R^2}.\n\\end{equation}\nSo if $P_c=-\\rho_c$, we obtain\n\\begin{equation}\n   m=0, \\text{independently of}\\quad k.\n\\end{equation}\nApplied to Eq. (\\ref{E:NEC}), the NEC\nis violated if\n\\begin{equation}\n   k=0\\quad \\text{or}\\quad k=-1.\n\\end{equation}\nThese conditions correspond to an open\nUniverse.\n\nTo summarize, we employed basic physical\nprinciples to derive the following\nzero-tidal force solution:\n\\begin{equation}\n   \\Phi(r)\\equiv 0\n\\end{equation}\nand (since $m=0$)\n\\begin{multline}\\label{E:shape1}\n  b(r)=\\frac{4M\\sqrt{\\beta}}{\\pi}\n  \\left(\\frac{1}{\\sqrt{\\beta}}\\text{tan}^{-1}\n  \\frac{r}{\\sqrt{\\beta}}-\\frac{r}{r^2+\\beta}\n  \\right)\\\\\n  -\\frac{4M\\sqrt{\\beta}}{\\pi}\n  \\left(\\frac{1}{\\sqrt{\\beta}}\\text{tan}^{-1}\n  \\frac{r_0}{\\sqrt{\\beta}}-\\frac{r_0}{r_0^2\n  +\\beta}\\right)+r_0.\n\\end{multline}\nThe slowly evolving wormhole solution is\nrestricted to the values $k=0$ and $k=-1$\nto ensure that the NEC is violated.  The\nwormhole spacetime is asymptotically flat.\n\n\\section{The special case $k=0$}\n    \\label{S:special}\nFor completeness let us briefly consider\na wormhole solution that does not depend\non the separation of the Einstein field\nequations.  We can combine Eqs.\n(\\ref{E:E1}) and (\\ref{E:E2})\nto obtain\n\\begin{equation}\n   8\\pi r^3R^2\\left[\\rho(r,t)+P^r(r,t)\n   \\right]=2r^3(\\dot{R}^2-R\\ddot{R})\n   +2r^3k+rb'(r)-b(r).\n\\end{equation}\nIf we now let $k=0$, then Eq.\n(\\ref{E:line2}) represents an evolving\nMorris-Thorne wormhole with the usual\nshape function $b=b(r)$.  The NEC is\nviolated at the throat $r=r_0$ for all\n$t$ whenever\n\\begin{equation}\\label{E:NEC1}\n   8\\pi r_0^3R^2\\left[\\rho(r_0,t)-\n   P^r(r_0,t)\\right]=2r_0^3(\\dot{R^2}\n   -R\\ddot{R})+r_0b'(r_0)-b(r_0)<0.\n\\end{equation}\nIf the Universe is indeed accelerating,\nthen the term $-R\\ddot{R}$ eventually\nbecomes dominant due to the\never-increasing $R$.  So for\nsufficiently large $R$, the NEC is\nviolated, thereby fulfilling a key\nrequirement for the existence of\nwormholes.  (Inflating Lorentzian\nwormholes are discussed in Ref.\n\\cite{tR93}.)\n\nRecalling that the radial tension $\\tau$\nis the negative of $P^r$, Inequality\n(\\ref{E:NEC1}) can be written (since\n$b(r_0)=r_0$)\n\\begin{equation}\\label{E:NEC2}\n   8\\pi r_0^2R^2\\left[\\tau(r_0)-\\rho(r_0)\n   \\right]=2r_0^2(-\\dot{R^2}+R\\ddot{R})\n   -b'(r_0)+1>0.\n\\end{equation}\nIf $R(t)\\equiv 1$, this reduces to the\nstatic Morris-Thorne wormhole; so if\n$b'(r_0)<1$, then $\\tau(r_0)>\\rho(r_0)$,\nrequiring exotic matter.  In Inequality\n(\\ref{E:NEC2}), however, $\\tau(r_0)>\n\\rho(r_0)$ could result from the\ndominant term $R\\ddot{R}$.  In that\ncase, the NEC is violated without\nrequiring exotic matter  for the\nconstruction of the wormhole itself.\n\n\\section{Comparison to an earlier solution}\n    \\label{S:comparison}\nA wormhole solution inspired by\nnoncommutative geometry had already been\nconsidered in Ref. \\cite{pK15}.  The\nEinstein field equation $\\rho(r)=\nb'(r)\/(8\\pi r^2)$, together with Eq.\n(\\ref{E:rho}), leads directly to the\nstatic solution, Eq. (\\ref{E:shape1}).\n(Here it is understood that $k=0$,\nbut $R(t)$ could be retained.)\nUnfortunately, this simple approach\nleaves the redshift function\nundetermined.  The desirability of\nzero tidal forces then suggested the\nassumption $\\Phi(r)\\equiv 0$ in Ref.\n\\cite{pK15}.  It is shown in Ref.\n\\cite{pK09}, however, that this\nassumption causes a Morris-Thorne\nwormhole to be incompatible with the\nFord-Roman constraints from quantum\nfield theory.  Given the\nnoncommutative-geometry background,\nrather than the purely classical setting\nin Ref. \\cite{MT88}, this objection\ndoes not apply directly.\n\nIt is interesting to note that in the\npresent paper, the zero-tidal force\nsolution is built into the Sung-Won\nKim model and does not require any\nadditional considerations.\n\n\\section{Conclusion}\\label{S:conclusion}\nMorris-Thorne wormholes typically require\na reverse strategy for their theoretical\nconstruction: specify the geometric\nrequirements and then manufacture or search\nthe Universe for matter or fields to obtain\nthe required energy-momentum tensor.  One\nof the goals in this paper is to obtain a\ncomplete wormhole solution from certain\nphysical principles.  To this end, we assume\na noncommutative-geometry background, as in\nprevious studies, but we also depend on a\ncosmological model due to Sung-Won Kim that\nis based on the FLRW model with a\ntraversable wormhole.  The basic assumption\nis that the matter content can be divided\ninto two parts, a cosmological part that\ndepends only on $t$ and a wormhole part\nthat depends only on the radial coordinate\n$r$.  The result is a complete zero-tidal\nforce solution; it is restricted, however,\nto the values $k=0$ and $k=-1$, corresponding\nto an open Universe.  This conclusion is\nconsistent with the special case $k=0$\ndiscussed in Sections \\ref{S:special}\nand \\ref{S:comparison}.\n\n\nThe wormhole is slowly evolving due to the\nscale factor $R(t)$ and, critically, the\nnoncommutative-geometry background not\nonly produces the wormhole solution, it\nalso affects in a direct manner the\ncosmological part of the solution.  This\nconclusion differs significantly from\nthose in Refs. \\cite {Kim} and \\cite{mC13}.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe inequalit\n\\begin{equation}\nf\\left( \\frac{a+b}{2}\\right) \\leq \\frac{1}{b-a}\\int_{a}^{b}f\\left( x\\right)\ndx\\leq \\frac{f\\left( a\\right) +f\\left( b\\right) }{2}  \\label{h}\n\\end{equation\nwhich holds for all convex functions $f:[a,b]\\rightarrow \n\\mathbb{R}\n$, is known in the literature as Hermite-Hadamard's inequality.\n\nIn \\cite{G}, Toader defined $m-$convexity as the following:\n\n\\begin{definition}\nThe function $f:[0,b]\\rightarrow \n\\mathbb{R}\n,$ $b>0$, is said to be $m-$convex where $m\\in \\lbrack 0,1],$ if we hav\n\\begin{equation*}\nf(tx+m(1-t)y)\\leq tf(x)+m(1-t)f(y)\n\\end{equation*\nfor all $x,y\\in \\lbrack 0,b]$ and $t\\in \\lbrack 0,1].$ We say that $f$ is \nm- $concave if $\\left( -f\\right) $ is $m-$convex.\n\\end{definition}\n\nIn \\cite{D}, Dragomir proved the following theorem.\n\nLet $f:\\left[ 0,\\infty \\right) \\rightarrow \n\\mathbb{R}\n$ be an $m-$convex function with $m\\in \\left( 0,1\\right] $ and $0\\leq a<b.$\nIf $f\\in L_{1}\\left[ a,b\\right] ,$ then the following inequalities hold\n\\begin{eqnarray}\nf\\left( \\frac{a+b}{2}\\right) &\\leq &\\frac{1}{b-a}\\int_{a}^{b}\\frac{f\\left(\nx\\right) +mf\\left( \\frac{x}{m}\\right) }{2}dx  \\label{h1} \\\\\n&\\leq &\\frac{1}{2}\\left[ \\frac{f\\left( a\\right) +mf\\left( \\dfrac{a}{m\n\\right) }{2}+m\\frac{f\\left( \\frac{b}{m}\\right) +mf\\left( \\dfrac{b}{m^{2}\n\\right) }{2}\\right] .  \\notag\n\\end{eqnarray}\n\nIn \\cite{DI} and \\cite{DPP}, the authors connect together some disparate\nthreads through a Hermite-Hadamard motif. The first of these threads is the\nunifying concept of a $g-$convex dominated function. Similarly, in \\cite{KOS\n, Kavurmac\\i\\ et al. introduced the following class of functions and then\nproved a theorem for this class of functions related to (\\ref{h1}).\n\n\\begin{definition}\nLet $g:\\left[ 0,b\\right] \\rightarrow \n\\mathbb{R}\n$ be a given $m-$convex function on the interval $\\left[ 0,b\\right] $. The\nreal function $f:\\left[ 0,b\\right] \\rightarrow \n\\mathbb{R}\n$ is called $\\left( g,m\\right) -$convex dominated on $\\left[ 0,b\\right] $ if\nthe following condition is satisfie\n\\begin{eqnarray*}\n&&\\left\\vert \\lambda f(x)+m(1-\\lambda )f(y)-f\\left( \\lambda x+m\\left(\n1-\\lambda \\right) y\\right) \\right\\vert \\\\\n&\\leq &\\lambda g(x)+m(1-\\lambda )g(y)-g\\left( \\lambda x+m\\left( 1-\\lambda\n\\right) y\\right)\n\\end{eqnarray*\nfor all $x,y\\in \\left[ 0,b\\right] $, $\\lambda \\in \\left[ 0,1\\right] $ and \nm\\in \\left[ 0,1\\right] .$\n\\end{definition}\n\n\\begin{theorem}\n\\label{a} Let $g:\\left[ 0,\\infty \\right) \\rightarrow \n\\mathbb{R}\n$ be an $m-$convex function with $m\\in \\left( 0,1\\right] $. $f:\\left[\n0,\\infty \\right) \\rightarrow \n\\mathbb{R}\n$ is $\\left( g,m\\right) -$convex dominated \\ mapping and $0\\leq a<b.$ If \nf\\in L_{1}\\left[ a,b\\right] ,$ then one has the inequalities\n\\begin{eqnarray*}\n&&\\left\\vert \\frac{1}{b-a}\\int_{a}^{b}\\frac{f\\left( x\\right) +mf\\left( \\frac\nx}{m}\\right) }{2}dx-f\\left( \\frac{a+b}{2}\\right) \\right\\vert \\\\\n&& \\\\\n&\\leq &\\frac{1}{b-a}\\int_{a}^{b}\\frac{g\\left( x\\right) +mg\\left( \\frac{x}{m\n\\right) }{2}dx-g\\left( \\frac{a+b}{2}\\right)\n\\end{eqnarray*\nan\n\\begin{eqnarray*}\n&&\\left\\vert \\frac{1}{2}\\left[ \\frac{f\\left( a\\right) +mf\\left( \\dfrac{a}{m\n\\right) }{2}+m\\frac{f\\left( \\frac{b}{m}\\right) +mf\\left( \\dfrac{b}{m^{2}\n\\right) }{2}\\right] -\\frac{1}{b-a}\\int_{a}^{b}\\frac{f\\left( x\\right)\n+mf\\left( \\frac{x}{m}\\right) }{2}dx\\right\\vert \\\\\n&& \\\\\n&\\leq &\\frac{1}{2}\\left[ \\frac{g\\left( a\\right) +mg\\left( \\dfrac{a}{m\n\\right) }{2}+m\\frac{g\\left( \\frac{b}{m}\\right) +mg\\left( \\dfrac{b}{m^{2}\n\\right) }{2}\\right] -\\frac{1}{b-a}\\int_{a}^{b}\\frac{g\\left( x\\right)\n+mg\\left( \\frac{x}{m}\\right) }{2}dx.\n\\end{eqnarray*}\n\\end{theorem}\n\nIn \\cite{MIH}, definition of $\\left( \\alpha ,m\\right) -$convexity was\nintroduced by Mihe\\c{s}an as the following.\n\n\\begin{definition}\n\\label{d1} The function $f:[0,b]\\rightarrow \n\\mathbb{R}\n,$ $b>0$, is said to be $\\left( \\alpha ,m\\right) -$convex, where $\\left(\n\\alpha ,m\\right) \\in \\lbrack 0,1]^{2},$ if we hav\n\\begin{equation*}\nf(tx+m(1-t)y)\\leq t^{\\alpha }f(x)+m(1-t^{\\alpha })f(y)\n\\end{equation*\nfor all $x,y\\in \\lbrack 0,b]$ and $t\\in \\lbrack 0,1].$\n\\end{definition}\n\nDenote by $K_{m}^{\\alpha }\\left( b\\right) $ the class of all $\\left( \\alpha\n,m\\right) -$convex functions on $\\left[ 0,b\\right] $ for which $f\\left(\n0\\right) \\leq 0.$ If we take $\\left( \\alpha ,m\\right) =\\left\\{ \\left(\n0,0\\right) ,\\left( \\alpha ,0\\right) ,\\left( 1,0\\right) ,\\left( 1,m\\right)\n,\\left( 1,1\\right) ,\\left( \\alpha ,1\\right) \\right\\} ,$ it can be easily\nseen that $\\left( \\alpha ,m\\right) -$convexity reduces to increasing: \n\\alpha -$starshaped, starshaped, $m-$convex, convex and $\\alpha -$convex,\nrespectively.\n\nIn \\cite{SSOR}, Set et al. proved the following Hadamard type inequalities\nfor $\\left( \\alpha ,m\\right) -$convex functions.\n\n\\begin{theorem}\nLet $f:\\left[ 0,\\infty \\right) \\rightarrow \n\\mathbb{R}\n$ be an $\\left( \\alpha ,m\\right) -$convex function with $\\left( \\alpha\n,m\\right) \\in \\left( 0,1\\right] ^{2}.$ If \\ $0\\leq a<b<\\infty $ and $f\\in $ \nL_{1}\\left[ a,b\\right] \\cap L_{1}\\left[ \\frac{a}{m},\\frac{b}{m}\\right] ,$\nthen the following inequality holds\n\\begin{equation}\nf\\left( \\frac{a+b}{2}\\right) \\leq \\frac{1}{b-a}\\int_{a}^{b}\\frac{f\\left(\nx\\right) +m\\left( 2^{\\alpha }-1\\right) f\\left( \\frac{x}{m}\\right) }\n2^{\\alpha }}dx.  \\label{h2}\n\\end{equation}\n\\end{theorem}\n\n\\begin{theorem}\nLet $f:\\left[ 0,\\infty \\right) \\rightarrow \n\\mathbb{R}\n$ be an $\\left( \\alpha ,m\\right) -$convex function with $\\left( \\alpha\n,m\\right) \\in \\left( 0,1\\right] ^{2}.$ If \\ $0\\leq a<b<\\infty $ and $f\\in $ \nL_{1}\\left[ a,b\\right] ,$ then the following inequality holds\n\\begin{equation}\n\\frac{1}{b-a}\\int_{a}^{b}f\\left( x\\right) dx\\leq \\frac{1}{2}\\left[ \\frac\nf\\left( a\\right) +f\\left( b\\right) +m\\alpha f\\left( \\frac{a}{m}\\right)\n+m\\alpha f\\left( \\frac{b}{m}\\right) }{\\alpha +1}\\right] .  \\label{h3}\n\\end{equation}\n\\end{theorem}\n\nFor the recent results based on the above definition see the papers \\cit\n{BOP}, \\cite{BPR}, \\cite{OKS}, \\cite{OAK} and \\cite{SSO}.\n\nIn \\cite{OSA}, the power mean $M_{r}(x,y;\\lambda )$ of order $r$ of positive\nnumbers $x,y$ is defined b\n\\begin{equation*}\nM_{r}(x,y;\\lambda )=\\left\\{ \n\\begin{array}{cc}\n\\left( \\lambda x^{r}+\\left( 1-\\lambda \\right) y^{r}\\right) ^{\\frac{1}{r}}, & \nr\\neq 0 \\\\ \nx^{\\lambda }y^{1-\\lambda }, & r=0\n\\end{array\n\\right. \n\\end{equation*\nA positive function $f$ is $r-$convex on $[a,b]$ if for all $x,y\\in \\lbrack\na,b]$ and $\\lambda \\in \\lbrack 0,1]\n\\begin{equation}\nf(\\lambda x+(1-\\lambda )y)\\leq M_{r}(f\\left( x\\right) ,f\\left( y\\right)\n;\\lambda ).  \\label{h4}\n\\end{equation\nThe generalized logarithmic mean of order $r$ of positive numbers $x,y$ is\ndefined b\n\\begin{equation}\nL_{r}\\left( x,y\\right) =\\left\\{ \n\\begin{array}{cc}\n\\frac{r}{r+1}\\frac{x^{r+1}-y^{r+1}}{x^{r}-y^{r}}, & r\\neq 0,1,x\\neq y \\\\ \n&  \\\\ \n\\frac{x-y}{\\ln x-\\ln y}, & r=0,x\\neq y \\\\ \n&  \\\\ \nxy\\frac{\\ln x-\\ln y}{x-y}, & r=-1,x\\neq y \\\\ \n&  \\\\ \nx, & x=\n\\end{array\n\\right.   \\label{L}\n\\end{equation\nIn \\cite{GPP}, the following theorem was proved by Gill et al. for $r-\nconvex functions.\n\n\\begin{theorem}\nSuppose $f$ is a positive $r-$convex function on $\\left[ a,b\\right] .$ Then \n\\begin{equation}\n\\frac{1}{b-a}\\int_{a}^{b}f\\left( x\\right) dx\\leq L_{r}\\left( f\\left(\na\\right) ,f\\left( b\\right) \\right) .  \\label{h5}\n\\end{equation\nIf $f$ is a positive $r-$concave function, then the inequality is reversed.\n\\end{theorem}\n\nIn the following sections our main results are given: We establish several\nnew convex dominated functions and then we obtain new Hadamard type\ninequalities.\n\n\\section{$\\left( g-\\left( \\protect\\alpha ,m\\right) \\right) $-convex\ndominated functions}\n\n\\begin{definition}\n\\label{d2} Let $g:\\left[ 0,b\\right] \\rightarrow \n\\mathbb{R}\n,$ $b>0$ be a given $\\left( \\alpha ,m\\right) $-convex function on the\ninterval $\\left[ 0,b\\right] $. The real function $f:\\left[ 0,b\\right]\n\\rightarrow \n\\mathbb{R}\n$ is called $\\left( g-\\left( \\alpha ,m\\right) \\right) $-convex dominated on \n\\left[ 0,b\\right] $ if the following condition is satisfie\n\\begin{eqnarray}\n&&\\left\\vert \\lambda ^{\\alpha }f(x)+m(1-\\lambda ^{\\alpha })f(y)-f\\left(\n\\lambda x+m\\left( 1-\\lambda \\right) y\\right) \\right\\vert  \\label{h6} \\\\\n&\\leq &\\lambda ^{\\alpha }g(x)+m(1-\\lambda ^{\\alpha })g(y)-g\\left( \\lambda\nx+m\\left( 1-\\lambda \\right) y\\right)  \\notag\n\\end{eqnarray\nfor all $x,y\\in \\left[ 0,b\\right] $, $\\lambda \\in \\left[ 0,1\\right] $ and \n\\left( \\alpha ,m\\right) \\in \\left[ 0,1\\right] ^{2}.$\n\\end{definition}\n\nThe next simple characterisation of $\\left( \\alpha ,m\\right) $-convex\ndominated functions holds.\n\n\\begin{lemma}\n\\label{l1} Let $g:\\left[ 0,b\\right] \\rightarrow \n\\mathbb{R}\n$ be an $\\left( \\alpha ,m\\right) $-convex function on the interval $\\left[\n0,b\\right] $ and the function $f:\\left[ 0,b\\right] \\rightarrow \n\\mathbb{R}\n.$ The following statements are equivalent:\n\\end{lemma}\n\n\\begin{enumerate}\n\\item $f$ is $\\left( g-\\left( \\alpha ,m\\right) \\right) $-convex dominated on \n$\\left[ 0,b\\right] .$\n\n\\item The mappings $g-f$ and $g+f$ are $\\left( \\alpha ,m\\right) $-convex\nfunctions on $\\left[ 0,b\\right] .$\n\n\\item There exist two $\\left( \\alpha ,m\\right) $-convex mappings $h,k$\ndefined on $\\left[ 0,b\\right] $ such tha\n\\begin{equation*}\n\\begin{array}{ccc}\nf=\\frac{1}{2}\\left( h-k\\right) & \\text{and} & g=\\frac{1}{2}\\left( h+k\\right\n\\end{array\n.\n\\end{equation*}\n\\end{enumerate}\n\n\\begin{proof}\n1$\\Longleftrightarrow $2 The condition (\\ref{h6}) is equivalent t\n\\begin{eqnarray*}\n&&g\\left( \\lambda x+m\\left( 1-\\lambda \\right) y\\right) -\\lambda ^{\\alpha\n}g(x)-m(1-\\lambda ^{\\alpha })g(y) \\\\\n&\\leq &\\lambda ^{\\alpha }f(x)+m(1-\\lambda ^{\\alpha })f(y)-f\\left( \\lambda\nx+m\\left( 1-\\lambda \\right) y\\right)  \\\\\n&\\leq &\\lambda ^{\\alpha }g(x)+m(1-\\lambda ^{\\alpha })g(y)-g\\left( \\lambda\nx+m\\left( 1-\\lambda \\right) y\\right) \n\\end{eqnarray*\nfor all $x,y\\in I$, $\\lambda \\in \\left[ 0,1\\right] $ and $\\left( \\alpha\n,m\\right) \\in \\left[ 0,1\\right] ^{2}.$ The two inequalities may be\nrearranged a\n\\begin{equation*}\n\\left( g+f\\right) \\left( \\lambda x+m\\left( 1-\\lambda \\right) y\\right) \\leq\n\\lambda ^{\\alpha }\\left( g+f\\right) (x)+m(1-\\lambda ^{\\alpha })\\left(\ng+f\\right) (y)\n\\end{equation*\nan\n\\begin{equation*}\n\\left( g-f\\right) \\left( \\lambda x+m\\left( 1-\\lambda \\right) y\\right) \\leq\n\\lambda ^{\\alpha }\\left( g-f\\right) (x)+m(1-\\lambda ^{\\alpha })\\left(\ng-f\\right) (y)\n\\end{equation*\nwhich are equivalent to the $\\left( \\alpha ,m\\right) $-convexity of $g+f$\nand $g-f,$ respectively.\n\n2$\\Longleftrightarrow $3 We define the mappings $f,g$ as $f=\\frac{1}{2\n\\left( h-k\\right) $ and $g=\\frac{1}{2}\\left( h+k\\right) $. Then, if we sum\nand subtract $f,g,$ respectively, we have $g+f=h$ and $g-f=k.$ By the\ncondition 2 of Lemma 1, the mappings $g-f$ and $g+f$ are $\\left( \\alpha\n,m\\right) $-convex on $\\left[ 0,b\\right] ,$ so $h,k$ are $\\left( \\alpha\n,m\\right) $-convex mappings too.\n\\end{proof}\n\n\\begin{theorem}\n\\label{t1} Let $g:\\left[ 0,\\infty \\right) \\rightarrow \n\\mathbb{R}\n$ be an $\\left( \\alpha ,m\\right) -$convex function with $\\left( \\alpha\n,m\\right) \\in \\left( 0,1\\right] ^{2}$. $f:\\left[ 0,\\infty \\right)\n\\rightarrow \n\\mathbb{R}\n$ is $\\left( g-\\left( \\alpha ,m\\right) \\right) -$convex dominated mapping\nand $0\\leq a<b.$ If $f\\in L_{1}\\left[ a,b\\right] \\cap L_{1}\\left[ \\frac{a}{m\n,\\frac{b}{m}\\right] ,$ then the first inequality holds\n\\begin{eqnarray*}\n&&\\left\\vert \\frac{1}{b-a}\\int_{a}^{b}\\frac{f\\left( x\\right) +m\\left(\n2^{\\alpha }-1\\right) f\\left( \\frac{x}{m}\\right) }{2^{\\alpha }}dx-f\\left( \n\\frac{a+b}{2}\\right) \\right\\vert \\\\\n&& \\\\\n&\\leq &\\frac{1}{b-a}\\int_{a}^{b}\\frac{g\\left( x\\right) +m\\left( 2^{\\alpha\n}-1\\right) g\\left( \\frac{x}{m}\\right) }{2^{\\alpha }}dx-g\\left( \\frac{a+b}{2\n\\right)\n\\end{eqnarray*\nand if $f\\in L_{1}\\left[ a,b\\right] $ then the second inequality holds: \n\\begin{eqnarray*}\n&&\\left\\vert \\frac{1}{2}\\left[ \\frac{f\\left( a\\right) +mf\\left( \\dfrac{a}{m\n\\right) }{\\alpha +1}+m\\alpha \\frac{f\\left( \\frac{b}{m}\\right) +mf\\left( \n\\dfrac{b}{m^{2}}\\right) }{\\alpha +1}\\right] -\\frac{1}{b-a}\\int_{a}^{b}\\frac\nf\\left( x\\right) +mf\\left( \\frac{x}{m}\\right) }{2}dx\\right\\vert \\\\\n&& \\\\\n&\\leq &\\frac{1}{2}\\left[ \\frac{g\\left( a\\right) +mg\\left( \\dfrac{a}{m\n\\right) }{\\alpha +1}+m\\alpha \\frac{g\\left( \\frac{b}{m}\\right) +mg\\left( \n\\dfrac{b}{m^{2}}\\right) }{\\alpha +1}\\right] -\\frac{1}{b-a}\\int_{a}^{b}\\frac\ng\\left( x\\right) +mg\\left( \\frac{x}{m}\\right) }{2}dx.\n\\end{eqnarray*}\n\\end{theorem}\n\n\\begin{proof}\nBy Definition \\ref{d2} with $\\lambda =\\frac{1}{2}$, as the mapping $f$ is \n\\left( g-\\left( \\alpha ,m\\right) \\right) -$convex dominated function, we\nhave tha\n\\begin{equation*}\n\\left\\vert \\frac{f\\left( x\\right) +m\\left( 2^{\\alpha }-1\\right) f\\left(\ny\\right) }{2^{\\alpha }}-f\\left( \\frac{x+my}{2}\\right) \\right\\vert \\leq \\frac\ng\\left( x\\right) +m\\left( 2^{\\alpha }-1\\right) g\\left( y\\right) }{2^{\\alpha \n}-g\\left( \\frac{x+my}{2}\\right)\n\\end{equation*\nfor all $x,y\\in \\left[ 0,\\infty \\right) $ and $\\left( \\alpha ,m\\right) \\in\n\\left( 0,1\\right] ^{2}.$ If we choose $x=ta+(1-t)b,$ $y=\\left( 1-t\\right) \n\\frac{a}{m}+t\\frac{b}{m}$ and $t\\in \\left[ 0,1\\right] ,$ then we ge\n\\begin{eqnarray*}\n&&\\left\\vert \\frac{f\\left( ta+(1-t)b\\right) +m\\left( 2^{\\alpha }-1\\right)\nf\\left( \\frac{\\left( 1-t\\right) a+tb}{m}\\right) }{2^{\\alpha }}-f\\left( \\frac\na+b}{2}\\right) \\right\\vert \\\\\n&& \\\\\n&\\leq &\\frac{g\\left( ta+(1-t)b\\right) +m\\left( 2^{\\alpha }-1\\right) g\\left( \n\\frac{\\left( 1-t\\right) a+tb}{2}\\right) }{2^{\\alpha }}-g\\left( \\frac{a+b}{2\n\\right) .\n\\end{eqnarray*\nIntegrating over $t$ on $\\left[ 0,1\\right] $ we deduce tha\n\\begin{eqnarray*}\n&&\\left\\vert \\frac{\\int_{0}^{1}f\\left( ta+(1-t)b\\right) dt+m\\left( 2^{\\alpha\n}-1\\right) \\int_{0}^{1}f\\left( \\frac{\\left( 1-t\\right) a+tb}{m}\\right) dt}\n2^{\\alpha }}-f\\left( \\frac{a+b}{2}\\right) \\right\\vert \\\\\n&& \\\\\n&\\leq &\\frac{\\int_{0}^{1}g\\left( ta+(1-t)b\\right) dt+m\\left( 2^{\\alpha\n}-1\\right) \\int_{0}^{1}g\\left( \\frac{\\left( 1-t\\right) a+tb}{m}\\right) dt}\n2^{\\alpha }}-g\\left( \\frac{a+b}{2}\\right)\n\\end{eqnarray*\nand so the first inequality is proved.\n\nSince $f$ is $\\left( g-\\left( \\alpha ,m\\right) \\right) -$convex dominated\nfunction, we hav\n\\begin{eqnarray*}\n&&\\left\\vert t^{\\alpha }f\\left( x\\right) +m(1-t^{\\alpha })f\\left( y\\right)\n-f\\left( tx+m(1-t)y\\right) \\right\\vert \\\\\n&& \\\\\n&\\leq &t^{\\alpha }g\\left( x\\right) +m(1-t^{\\alpha })g\\left( y\\right)\n-g\\left( tx+m(1-t)y\\right) ,\\text{ for all }x,y>0\n\\end{eqnarray*\nwhich gives for $x=a$ and $y=\\frac{b}{m}\n\\begin{eqnarray}\n&&\\left\\vert t^{\\alpha }f\\left( a\\right) +m(1-t^{\\alpha })f\\left( \\frac{b}{m\n\\right) -f\\left( ta+m(1-t)\\frac{b}{m}\\right) \\right\\vert  \\label{h7} \\\\\n&&  \\notag \\\\\n&\\leq &t^{\\alpha }g\\left( a\\right) +m(1-t^{\\alpha })g\\left( \\frac{b}{m\n\\right) -g\\left( ta+m(1-t)\\frac{b}{m}\\right)  \\notag\n\\end{eqnarray\nand for $x=\\frac{a}{m}$, $y=\\frac{b}{m^{2}}$ and then multiply with $m$ \n\\begin{eqnarray}\n&&\\left\\vert mtf\\left( \\frac{a}{m}\\right) +m^{2}(1-t)f\\left( \\frac{b}{m^{2}\n\\right) -mf\\left( t\\frac{a}{m}+(1-t)\\frac{b}{m}\\right) \\right\\vert\n\\label{h8} \\\\\n&&  \\notag \\\\\n&\\leq &mtg\\left( \\frac{a}{m}\\right) +m^{2}(1-t)g\\left( \\frac{b}{m^{2}\n\\right) -mg\\left( t\\frac{a}{m}+(1-t)\\frac{b}{m}\\right)  \\notag\n\\end{eqnarray\nfor all $t\\in \\left[ 0,1\\right] .$ By properties of modulus, if we add the\ninequalities in $\\left( \\text{\\ref{h7}}\\right) $ and $\\left( \\text{\\ref{h8}\n\\right) $, we get \n\\begin{eqnarray*}\n&&\\left\\vert t^{\\alpha }\\left[ f\\left( a\\right) +mf\\left( \\frac{a}{m}\\right)\n\\right] +m(1-t^{\\alpha })\\left[ f\\left( \\frac{b}{m}\\right) +mf\\left( \\frac{\n}{m^{2}}\\right) \\right] \\right. \\\\\n&& \\\\\n&&-\\left. \\left[ f\\left( ta+m(1-t)\\frac{b}{m}\\right) +mf\\left( t\\frac{a}{m\n+(1-t)\\frac{b}{m}\\right) \\right] \\right\\vert \\\\\n&& \\\\\n&\\leq &t^{\\alpha }\\left[ g\\left( a\\right) +mg\\left( \\frac{a}{m}\\right)\n\\right] +m(1-t^{\\alpha })\\left[ g\\left( \\frac{b}{m}\\right) +mg\\left( \\frac{\n}{m^{2}}\\right) \\right] \\\\\n&& \\\\\n&&-\\left[ g\\left( ta+m(1-t)\\frac{b}{m}\\right) +mg\\left( t\\frac{a}{m}+(1-t\n\\frac{b}{m}\\right) \\right] .\n\\end{eqnarray*\nThus, integrating over $t$ on $\\left[ 0,1\\right] $ we obtain the second\ninequality. The proof is completed.\n\\end{proof}\n\n\\begin{remark}\nIf we choose $\\alpha =1$ in Theorem \\ref{t1}, we get two inequalities of\nHermite-Hadamard type for functions that are $\\left( g,m\\right) -$convex\ndominated in Theorem \\ref{a}.\n\\end{remark}\n\n\\begin{theorem}\n\\label{t2} Let $g:\\left[ 0,\\infty \\right) \\rightarrow \n\\mathbb{R}\n$ be an $\\left( \\alpha ,m\\right) -$convex function with $\\left( \\alpha\n,m\\right) \\in \\left( 0,1\\right] ^{2}$. $f:\\left[ 0,\\infty \\right)\n\\rightarrow \n\\mathbb{R}\n$ is $\\left( g-\\left( \\alpha ,m\\right) \\right) -$convex dominated \\ mapping\nand $0\\leq a<b.$ If $f\\in L_{1}\\left[ a,b\\right] ,$ then the following\ninequality holds\n\\begin{eqnarray}\n&&\\left\\vert \\frac{1}{2}\\left[ \\frac{f\\left( a\\right) +f\\left( b\\right)\n+m\\alpha f\\left( \\frac{a}{m}\\right) +m\\alpha f\\left( \\frac{b}{m}\\right) }\n\\alpha +1}\\right] -\\frac{1}{b-a}\\int_{a}^{b}f\\left( x\\right) dx\\right\\vert\n\\label{h9} \\\\\n&&  \\notag \\\\\n&\\leq &\\frac{1}{2}\\left[ \\frac{g\\left( a\\right) +g\\left( b\\right) +m\\alpha\ng\\left( \\frac{a}{m}\\right) +m\\alpha g\\left( \\frac{b}{m}\\right) }{\\alpha +1\n\\right] -\\frac{1}{b-a}\\int_{a}^{b}g\\left( x\\right) dx  \\notag\n\\end{eqnarray}\n\\end{theorem}\n\n\\begin{proof}\nSince $f$ is $\\left( g-\\left( \\alpha ,m\\right) \\right) -$convex dominated\nfunction, we hav\n\\begin{eqnarray*}\n&&\\left\\vert t^{\\alpha }f\\left( a\\right) +m(1-t^{\\alpha })f\\left( \\frac{b}{m\n\\right) -f\\left( ta+m(1-t)\\frac{b}{m}\\right) \\right\\vert \\\\\n&& \\\\\n&\\leq &t^{\\alpha }g\\left( a\\right) +m(1-t^{\\alpha })g\\left( \\frac{b}{m\n\\right) -g\\left( ta+m(1-t)\\frac{b}{m}\\right)\n\\end{eqnarray*\nan\n\\begin{eqnarray*}\n&&\\left\\vert t^{\\alpha }f\\left( b\\right) +m(1-t^{\\alpha })f\\left( \\frac{a}{m\n\\right) -f\\left( tb+m(1-t)\\frac{a}{m}\\right) \\right\\vert \\\\\n&& \\\\\n&\\leq &t^{\\alpha }g\\left( b\\right) +m(1-t^{\\alpha })g\\left( \\frac{a}{m\n\\right) -g\\left( tb+m(1-t)\\frac{a}{m}\\right)\n\\end{eqnarray*\nfor all $t\\in \\left[ 0,1\\right] .$ Adding the above inequalities, we ge\n\\begin{eqnarray*}\n&&\\left\\vert t^{\\alpha }\\left[ f\\left( a\\right) +f\\left( b\\right) \\right]\n+m(1-t^{\\alpha })\\left[ f\\left( \\frac{a}{m}\\right) +f\\left( \\frac{b}{m\n\\right) \\right] -f\\left( ta+m(1-t)\\frac{b}{m}\\right) -f\\left( tb+m(1-t)\\frac\na}{m}\\right) \\right\\vert \\\\\n&& \\\\\n&\\leq &t^{\\alpha }\\left[ g\\left( a\\right) +g\\left( b\\right) \\right]\n+m(1-t^{\\alpha })\\left[ g\\left( \\frac{a}{m}\\right) +g\\left( \\frac{b}{m\n\\right) \\right] -g\\left( ta+m(1-t)\\frac{b}{m}\\right) -g\\left( tb+m(1-t)\\frac\na}{m}\\right) .\n\\end{eqnarray*\nIntegrating over $t\\in \\left[ 0,1\\right] $ and then by dividing the\nresulting inequality with $2,$ we get the desired result. The proof is\ncompleted.\n\nAnother proof can be done as the following.\n\nSince $f$ is $\\left( g-\\left( \\alpha ,m\\right) \\right) -$convex dominated,\nwe have by Lemma \\ref{l1} that $g+f$ and $g-f$ are $\\left( \\alpha ,m\\right)\n- $convex on $\\left[ 0,b\\right] ,$ and so by the Hadamard's type inequality\nfor $\\left( \\alpha ,m\\right) -$convex functions in $\\left( \\text{\\ref{h3}\n\\right) \n\\begin{eqnarray}\n&&\\frac{1}{b-a}\\int_{a}^{b}\\left( g+f\\right) \\left( x\\right) dx  \\label{h10}\n\\\\\n&\\leq &\\frac{1}{2}\\left[ \\frac{\\left( g+f\\right) \\left( a\\right) +\\left(\ng+f\\right) \\left( b\\right) +m\\alpha \\left( g+f\\right) \\left( \\frac{a}{m\n\\right) +m\\alpha \\left( g+f\\right) \\left( \\frac{b}{m}\\right) }{\\alpha +1\n\\right]  \\notag\n\\end{eqnarray\nan\n\\begin{eqnarray}\n&&\\frac{1}{b-a}\\int_{a}^{b}\\left( g-f\\right) \\left( x\\right) dx  \\label{h11}\n\\\\\n&\\leq &\\frac{1}{2}\\left[ \\frac{\\left( g-f\\right) \\left( a\\right) +\\left(\ng-f\\right) \\left( b\\right) +m\\alpha \\left( g-f\\right) \\left( \\frac{a}{m\n\\right) +m\\alpha \\left( g-f\\right) \\left( \\frac{b}{m}\\right) }{\\alpha +1\n\\right]  \\notag\n\\end{eqnarray\nBy using the inequalities in $\\left( \\text{\\ref{h10}}\\right) $ and $\\left( \n\\text{\\ref{h11}}\\right) $, we get the inequality in $\\left( \\text{\\ref{h9}\n\\right) $.\n\\end{proof}\n\n\\section{$\\left( g,r\\right) -$convex dominated functions}\n\n\\begin{definition}\n\\label{d3} Let positive function $g:\\left[ a,b\\right] \\rightarrow \n\\mathbb{R}\n$ be a given $r-$convex function on $\\left[ a,b\\right] $. The real function \nf:\\left[ a,b\\right] \\rightarrow \n\\mathbb{R}\n$ is called $\\left( g,r\\right) -$convex dominated on $\\left[ a,b\\right] $ if\nthe following condition is satisfied\n\\begin{eqnarray*}\n&&\\left\\vert M_{r}(f\\left( x\\right) ,f\\left( y\\right) ;\\lambda )-f\\left(\n\\lambda x+\\left( 1-\\lambda \\right) y\\right) \\right\\vert  \\\\\n&\\leq &M_{r}(g\\left( x\\right) ,g\\left( y\\right) ;\\lambda )-g\\left( \\lambda\nx+\\left( 1-\\lambda \\right) y\\right) \n\\end{eqnarray*\nfor all $x,y\\in \\left[ a,b\\right] $ and $\\lambda \\in \\left[ 0,1\\right] .$\n\\end{definition}\n\n\\begin{theorem}\nLet positive function $g:\\left[ a,b\\right] \\rightarrow \n\\mathbb{R}\n$ be an $r-$convex function on $\\left[ a,b\\right] $. $f:\\left[ a,b\\right]\n\\rightarrow \n\\mathbb{R}\n$ is $\\left( g,r\\right) -$convex dominated\\ mapping and $0\\leq a<b.$ If \nf\\in L_{1}\\left[ a,b\\right] ,$ then the following inequality holds\n\\begin{equation*}\n\\left\\vert L_{r}\\left( f\\left( a\\right) ,f\\left( b\\right) \\right) -\\frac{1}\nb-a}\\int_{a}^{b}f\\left( x\\right) dx\\right\\vert \\leq L_{r}\\left( g\\left(\na\\right) ,g\\left( b\\right) \\right) -\\frac{1}{b-a}\\int_{a}^{b}g\\left(\nx\\right) dx\n\\end{equation*\nfor all $x,y\\in I$, $\\lambda \\in \\left[ 0,1\\right] $ and $L_{r}\\left(\nf\\left( a\\right) ,f\\left( b\\right) \\right) $ as in (\\ref{L}).\n\\end{theorem}\n\n\\begin{proof}\nBy the Definition \\ref{d3} with $r=0,\\ f\\left( a\\right) \\neq f\\left(\nb\\right) ,$ we have \n\\begin{eqnarray*}\n&&\\left\\vert f^{\\lambda }\\left( a\\right) f^{1-\\lambda }\\left( b\\right)\n-f\\left( \\lambda a+\\left( 1-\\lambda \\right) b\\right) \\right\\vert \\\\\n&\\leq &g^{\\lambda }\\left( a\\right) g^{1-\\lambda }\\left( b\\right) -g\\left(\n\\lambda a+\\left( 1-\\lambda \\right) b\\right) .\n\\end{eqnarray*\nIntegrating the above inequality over $\\lambda $ on $\\left[ 0,1\\right] ,$ we\nhave \n\\begin{eqnarray*}\n&&\\left\\vert f\\left( b\\right) \\int_{0}^{1}\\left[ \\frac{f\\left( a\\right) }\nf\\left( b\\right) }\\right] ^{\\lambda }d\\lambda -\\int_{0}^{1}f\\left( \\lambda\na+\\left( 1-\\lambda \\right) b\\right) d\\lambda \\right\\vert \\\\\n&& \\\\\n&\\leq &g\\left( b\\right) \\int_{0}^{1}\\left[ \\frac{g\\left( a\\right) }{g\\left(\nb\\right) }\\right] ^{\\lambda }d\\lambda -\\int_{0}^{1}g\\left( \\lambda a+\\left(\n1-\\lambda \\right) b\\right) d\\lambda .\n\\end{eqnarray*\nBy a simple calculation we hav\n\\begin{eqnarray*}\n&&\\left\\vert \\frac{f\\left( b\\right) -f\\left( a\\right) }{\\ln f\\left( b\\right)\n-\\ln f\\left( a\\right) }-\\frac{1}{b-a}\\int_{a}^{b}f\\left( x\\right)\ndx\\right\\vert \\\\\n&& \\\\\n&\\leq &\\frac{g\\left( b\\right) -g\\left( a\\right) }{\\ln g\\left( b\\right) -\\ln\ng\\left( a\\right) }-\\frac{1}{b-a}\\int_{a}^{b}g\\left( x\\right) dx.\n\\end{eqnarray*\nThe above inequality can written a\n\\begin{equation*}\n\\left\\vert L_{r}\\left( f\\left( a\\right) ,f\\left( b\\right) \\right) -\\frac{1}\nb-a}\\int_{a}^{b}f\\left( x\\right) dx\\right\\vert \\leq L_{r}\\left( g\\left(\na\\right) ,g\\left( b\\right) \\right) -\\frac{1}{b-a}\\int_{a}^{b}g\\left(\nx\\right) dx.\n\\end{equation*\nFor $r=0,\\ f\\left( a\\right) =f\\left( b\\right) ,$ we have with the same\ndevelopmen\n\\begin{eqnarray*}\n&&\\left\\vert f\\left( a\\right) -f\\left( \\lambda a+\\left( 1-\\lambda \\right)\nb\\right) \\right\\vert \\\\\n&\\leq &g\\left( a\\right) -g\\left( \\lambda a+\\left( 1-\\lambda \\right) b\\right)\n\\end{eqnarray*\nand this inequality can be written a\n\\begin{equation*}\n\\left\\vert L_{r}\\left( f\\left( a\\right) ,f\\left( b\\right) \\right) -\\frac{1}\nb-a}\\int_{a}^{b}f\\left( x\\right) dx\\right\\vert \\leq L_{r}\\left( g\\left(\na\\right) ,g\\left( b\\right) \\right) -\\frac{1}{b-a}\\int_{a}^{b}g\\left(\nx\\right) dx.\n\\end{equation*\nBy the Definition \\ref{d3} with $r\\neq 0,-1,\\ f\\left( a\\right) \\neq f\\left(\nb\\right) ,$ we hav\n\\begin{eqnarray*}\n&&\\left\\vert \\left( \\lambda f^{r}\\left( a\\right) +\\left( 1-\\lambda \\right)\nf^{r}\\left( b\\right) \\right) ^{\\frac{1}{r}}-f\\left( \\lambda a+\\left(\n1-\\lambda \\right) b\\right) \\right\\vert \\\\\n&& \\\\\n&\\leq &\\left( \\lambda g^{r}\\left( a\\right) +\\left( 1-\\lambda \\right)\ng^{r}\\left( b\\right) \\right) ^{\\frac{1}{r}}-g\\left( \\lambda a+\\left(\n1-\\lambda \\right) b\\right) .\n\\end{eqnarray*\nIntegrating the above inequality over $\\lambda $ on $\\left[ 0,1\\right] ,$ we\nhave \n\\begin{eqnarray*}\n&&\\left\\vert \\frac{r}{r+1}\\frac{f^{r+1}\\left( a\\right) -f^{r+1}\\left(\nb\\right) }{f^{r}\\left( a\\right) -f^{r}\\left( b\\right) }-\\frac{1}{b-a\n\\int_{a}^{b}f\\left( x\\right) dx\\right\\vert \\\\\n&& \\\\\n&\\leq &\\frac{r}{r+1}\\frac{g^{r+1}\\left( a\\right) -g^{r+1}\\left( b\\right) }\ng^{r}\\left( a\\right) -g^{r}\\left( b\\right) }-\\frac{1}{b-a\n\\int_{a}^{b}g\\left( x\\right) dx.\n\\end{eqnarray*\nThe above inequality can be written a\n\\begin{eqnarray*}\n&&\\left\\vert L_{r}\\left( f\\left( a\\right) ,f\\left( b\\right) \\right) -\\frac{\n}{b-a}\\int_{a}^{b}f\\left( x\\right) dx\\right\\vert \\\\\n&\\leq &L_{r}\\left( g\\left( a\\right) ,g\\left( b\\right) \\right) -\\frac{1}{b-a\n\\int_{a}^{b}g\\left( x\\right) dx.\n\\end{eqnarray*\nFor$\\ r\\neq 0$ and $f\\left( a\\right) =f\\left( b\\right) ,$ we have similarl\n\\begin{eqnarray*}\n&&\\left\\vert \\left( f^{r}\\left( a\\right) \\right) ^{\\frac{1}{r}}-f\\left(\n\\lambda a+\\left( 1-\\lambda \\right) b\\right) \\right\\vert \\\\\n&\\leq &\\left( g^{r}\\left( a\\right) \\right) ^{\\frac{1}{r}}-g\\left( \\lambda\na+\\left( 1-\\lambda \\right) b\\right) .\n\\end{eqnarray*\nThen integrating the above inequality over $\\lambda $ on $\\left[ 0,1\\right]\n, $ we hav\n\\begin{eqnarray*}\n&&\\left\\vert L_{r}\\left( f\\left( a\\right) ,f\\left( b\\right) \\right) -\\frac{\n}{b-a}\\int_{a}^{b}f\\left( x\\right) dx\\right\\vert \\\\\n&\\leq &L_{r}\\left( g\\left( a\\right) ,g\\left( b\\right) \\right) -\\frac{1}{b-a\n\\int_{a}^{b}g\\left( x\\right) dx.\n\\end{eqnarray*\nFinally, let $r=-1.$ For$\\ f\\left( a\\right) \\neq f\\left( b\\right) $ we have\nagai\n\\begin{eqnarray*}\n&&\\left\\vert \\left( \\lambda f^{-1}\\left( a\\right) +\\left( 1-\\lambda \\right)\nf^{-1}\\left( b\\right) \\right) ^{-1}-f\\left( \\lambda a+\\left( 1-\\lambda\n\\right) b\\right) \\right\\vert \\\\\n&& \\\\\n&\\leq &\\left( \\lambda g^{-1}\\left( a\\right) +\\left( 1-\\lambda \\right)\ng^{-1}\\left( b\\right) \\right) ^{-1}-g\\left( \\lambda a+\\left( 1-\\lambda\n\\right) b\\right) .\n\\end{eqnarray*\nIntegrating the above inequality over $\\lambda $ on $\\left[ 0,1\\right] ,$ we\nhave \n\\begin{eqnarray*}\n&&\\left\\vert \\frac{f\\left( a\\right) f\\left( b\\right) }{f\\left( b\\right)\n-f\\left( a\\right) }\\int_{\\frac{1}{f\\left( a\\right) }}^{\\frac{1}{f\\left(\nb\\right) }}\\lambda ^{-1}d\\lambda -\\frac{1}{b-a}\\int_{a}^{b}f\\left( x\\right)\ndx\\right\\vert \\\\\n&& \\\\\n&\\leq &\\frac{g\\left( a\\right) g\\left( b\\right) }{g\\left( b\\right) -g\\left(\na\\right) }\\int_{\\frac{1}{f\\left( a\\right) }}^{\\frac{1}{f\\left( b\\right) \n}\\lambda ^{-1}d\\lambda -\\frac{1}{b-a}\\int_{a}^{b}g\\left( x\\right) dx.\n\\end{eqnarray*\nThe above inequality can be written a\n\\begin{eqnarray*}\n&&\\left\\vert L_{-1}\\left( f\\left( a\\right) ,f\\left( b\\right) \\right) -\\frac{\n}{b-a}\\int_{a}^{b}f\\left( x\\right) dx\\right\\vert \\\\\n&\\leq &L_{-1}\\left( g\\left( a\\right) ,g\\left( b\\right) \\right) -\\frac{1}{b-a\n\\int_{a}^{b}g\\left( x\\right) dx.\n\\end{eqnarray*\nThe proof is completed.\n\\end{proof}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nPopulation estimation methods have been widely used in ecology to study the development of species over the last several decades~\\cite{krebs1989ecological}. Existing population estimation methods focus on open populations, where births, deaths, and migrations are taken into account, or closed population methods, where the population is assumed to remain static over the population estimation study. In this work, we focus on closed population methods and study how their respective population estimates can be used to evaluate an evaluation set of generated samples, given a reference set of real samples. In this work, samples are either contextualized word or sentence embeddings.\nWe study two mark-recapture methods, namely the Petersen~\\cite{petersen} and the Schnabel~\\cite{schnabel} estimators, where samples are captured and marked, released, and then recaptured. The number of marked, recaptured, and captured samples can then be used to estimate the population size. We additionally study one maximum-likelihood method, Program CAPTURE\n~\\cite{capture}, which uses the number of marked or unique samples over multiple captures to estimate the population size.\n\nAccurate evaluation of generated data is essential to correctly measure in what degrees we can improve the overall generation process. Depending on the use case, single-valued metrics may suffice to assess specific conditional language generation tasks, such as machine translation and text summarization, where we are interested in evaluating the similarities of a generated translation or summary to a specific reference translation or summary, respectively. \nOn the other hand, on unconditional language generation, for example, it may be useful to have separate measures for the diversity and quality of the generated set, enabling the identification of possible shortcomings of our generation system and try to fix it accordingly. This has been an active area of generative models research, specifically generative adversarial networks~\\cite{gans}, where several works have focused on stimulating diversity while maintaining the overall sample quality~\\cite{veegan,pacgan,dropoutgan,microbatchgan,sauder2020best}. Hence, depending on the context, a single-valued or double-valued metric may be more desirable.\n\nMark-Evaluate (ME) is a family of 3 novel language evaluation methods based on the above population estimation methods: ME$_\\text{Petersen}$ and ME$_\\text{CAPTURE}$ retrieve a single-valued metric to assess an evaluation set, while ME$_\\text{Schnabel}$ returns a double-valued metric, separately measuring the quality and diversity of the evaluation set.\nOur main contributions can be listed as follows:\n\\begin{enumerate*}[label=(\\roman*)]\n    \\item Proposal of 3 novel language metrics (Section~\\ref{sec:mark_evaluate}) that are sensitive to mode collapse (Section~\\ref{sec:mode_collapse}) and quality detriment (Section~\\ref{sec:swap}) and show a high correlation to human evaluation on challenging text generation tasks, such as unconditional language generation (Section~\\ref{sec:language_generation}), machine translation (Section~\\ref{sec:machine_translation}) and text summarization (Section~\\ref{sec:text_summarization}).\n    \\item In-depth study of the language assessment capability of popular existing metrics, \\textit{i.e.} FID~\\cite{fid}, PRD~\\cite{prd} and IMPAR~\\cite{impar}, primarily used to evaluate image generation in the past.\n    \\item Usage of contextual information and different levels of granularity to assess language, by using either contextualized word (Sections~\\ref{sec:machine_translation} and ~\\ref{sec:text_summarization}) or sentence embeddings (Sections~\\ref{sec:synthetic_experiments}, \\ref{sec:language_generation}) derived from BERT~\\cite{bert}.\n    \\item Code for the reproducibility of the results will be publicly available.\n\\end{enumerate*}\n\n\\section{Related work}\n\nWhile acknowledging the importance of traditional evaluation metrics, such as BLEU~\\cite{bleu}, ROUGE~\\cite{rouge} and METEOR~\\cite{meteor}, we will focus on the new trend of unsupervised methods that use embedding representations of pre-trained models to assess a set of evaluation samples. Our family of methods analyzes the data manifold to assess the evaluation set by using \\textit{k-nearest neighbors} to determine the capture volume. Several methods have been recently proposed to assess data generation using topological information, however, they were primarily intended to assess image generation~\\cite{prd,khrulkov2018geometry,impar,fti}.\nIn this work, we investigate the performance of such methods in the text domain, analyzing their behavior on synthetic experiments and their correlation with human evaluation.\n\nPrecision and recall for distributions (PRD) was proposed by~\\newcite{prd} and uses of \\textit{k-means}~\\cite{kmeans_2} to build histograms of the discrete reference and evaluation distributions over the clusters' centers. The evaluation distribution is then assessed in terms of relative probability densities. Precision is obtained by calculating the probability of an evaluation sample falling within the reference distribution's support. On the other hand, recall is retrieved by calculating the probability of a reference sample falling within the evaluation distribution's support.\n\n\\newcite{impar} suggested several improvements to the above method, which we call improved precision and recall (IMPAR). First, instead of \\textit{k-means}, they proposed to use \\textit{k-nearest neighbors} to approximate the reference and evaluation manifolds by building a hypersphere around each sample to its $k$-th nearest neighbor. Second, they simplify PRD's notions of precision and recall, by calculating the probability of an evaluation sample to fall within at least one reference sample's hypersphere, and vice-versa, respectively. The proposed manifold approximations by the usage of hyperspheres present a simple, yet effective way of representing the reference and evaluation manifold in an explicit, non-parametric way. We build upon this idea and use identical hyperspheres to determine the capture volume used by our different estimators to estimate the population size.\n\nFr\u00e9chet Inception Distance or FID~\\cite{fid} is a widely used single-valued metric that assesses data similarity by calculating the distance between the reference and evaluation distributions. Even though originally proposed for the image domain, \\newcite{semeniuta2018accurate} adapted FID to evaluate text generation by getting vector representations from InferSent~\\cite{infersent}, instead of Inception-V3~\\cite{inception}. Even though this metric precedes PRD and IMPAR, FID is still commonly used to assess generative models in the image and text domain.\n\nAs previously mentioned, we study both the usage of sentence embeddings, derived from SBERT~\\cite{sbert}, as well as contextualized word embeddings from BERT~\\cite{sbert}, which have been recently shown to improve language assessment, both in a supervised~\\cite{bertr,bleurt} and unsupervised manner~\\cite{moverscore,bertscore}.\nMore specifically, BERTScore measures precision and recall from a reference and evaluation text by calculating the required transport of each word of a given text to the most semantically similar word in the other text. On the other hand, MoverScore measures the semantic distance between two texts by calculating the minimum transport required between the reference and evaluation texts. \nThese metrics are ideal for conditional text generation, such as machine translation and text summarization, where an evaluation text should match a given reference text.\n\n\\section{Mark-Evaluate}\n\\label{sec:mark_evaluate}\n\nIn this work, we consider population estimation methods for closed populations, where the true population size remains constant throughout the estimation study. In our use case, our population consists of two sets, namely a reference set $S_r$ and an evaluation set $S_e$. The true population size ($P$) is then known \\textit{a priori} and represents the total number of samples in the two sets: $P=|S_r|+|S_e|$. Given an estimated population size ($\\widehat{P}$) from one of the used estimators, we measure the accuracy loss ($A$) as follows:\n\n\\begin{equation}\n\\label{eq:accuracy}\nA(P,\\widehat{P}) = \\text{max}\\Big(\\Big|\\frac{\\widehat{P}-P}{P}\\Big|,1\\Big),\n\\end{equation}\n\nwith a low accuracy loss, \\textit{i.e.} $A(P,\\widehat{P})\\approx 0$, representing a good population estimate, and a high accuracy loss, \\textit{i.e.} $A(P,\\widehat{P})\\approx 1$, otherwise.\nOur population estimation methods assume all samples to have an equal chance of capture, which is influenced by our capture volumes: hyperspheres that reaches each reference or evaluation sample's $k$-th nearest reference or evaluation neighbor, respectively.\nHence, if evaluation samples tend not to be inside any reference sample's hypersphere and vice-versa, the population estimate will likely be poor due to the lack of captured samples in the estimation study.\n\nOur methods can be separated into three categories: single marking and recapture (ME$_\\text{Petersen}$), multiple markings and recaptures (ME$_\\text{Schnabel}$) and multiple markings and captures (ME$_\\text{CAPTURE}$). Let us consider two sets of samples $S$ and $S'$, where each sample is a contextualized word embedding or a sentence embedding derived from BERT, depending on the task. \nFigure~\\ref{fig:mark_recapture} illustrates our family of methods. \n\n\\begin{figure}\n\\caption{Each sample $s \\in S$ and $s' \\in S'$ is represented as a blue and red circle, respectively. Marked samples are represented by filled circles. In this illustration, hyperspheres reach to each sample's nearest neighbor of the same set, \\textit{i.e.} $K=1$.\nFor ME$_\\text{Petersen}$ and ME$_\\text{Schnabel}$, we first capture and mark all samples inside any hypersphere of $s$. Then, for ME$_\\text{Petersen}$, we count the number of marked samples or recaptures inside any hypersphere of $s'$. For ME$_\\text{Schnabel}$, we perform a similar process iteratively, marking and recapturing samples inside the hypersphere of each $s'$, resulting in all samples being marked in the end.\nOn the other hand, ME$_\\text{CAPTURE}$ captures and marks samples inside each hypersphere of $s$ and $s'$.\n}\n\\centering\n\\includegraphics[width=0.9\\textwidth]{Figures\/method.pdf}\n\\label{fig:mark_recapture}\n\\end{figure}\n\nAdapting \\newcite{impar}'s formulations, we define a binary function $f$ that returns whether a sample $s' \\in S'$ lays inside any capture volume or hypersphere of a sample $s \\in S$:\n\n\\begin{equation}\n    f(s', S) = \n    \\begin{cases}\n        1, \\quad \\text{if} \\quad ||s'-s||_2 \\leq ||s - \\text{NN}_k(s, S)[-1]||_2 \\quad \\text{for at least one} \\quad s \\in S\\\\\n        0, \\quad \\text{otherwise,}\\\\\n    \\end{cases}\n\\end{equation}\n\nwhere NN$_k(s, S)$ returns an ordered set containing $s$ and its $k$-nearest neighbors in the set $S$, in ascending order of Euclidean distances to $s$. Hence, NN$_k(s, S)[-1]$ represents the $k$'th nearest neighbor of $s$.\nWe may refer to individual samples in $S$ and $S'$ as $\\{s_1,\\ldots,s_{|S|}\\}$ and $\\{s'_1,\\ldots,s'_{|S'|}\\}$, respectively.\n\nThe Petersen estimator~\\cite{petersen}, relies on a single marking step and a single recapture step. It merely assumes that the ratio of marked samples ($M$) in the marking step and the population size ($P$) is equivalent to the ratio of recaptured samples ($R$) and captured samples ($C$) in the recapture step. The population size estimate ($\\widehat{P}_\\text{Petersen}$) is then calculated as follows:\n\n\\begin{equation}\n\\label{eq:petersen}\n    \\widehat{P}_\\text{Petersen}(S, S') = \\dfrac{C(S, S') M(S, S')}{R(S, S')}.\n\\end{equation}\n\nDuring the marking step, we mark all samples inside at least one hypersphere of $s$: $M(S, S') = |S| + \\sum\\limits_{s' \\in S'} f(s', S)$. During the recapture step, we do the opposite, marking all samples inside at least one hypersphere of $s'$: $C(S, S') = |S'| + \\sum\\limits_{s \\in S} f(s, S')$. Additionally, in the recapture step, we count the number of captured samples that are already marked from the marking step, \\textit{i.e.} the recaptured samples. This corresponds to the number of samples in $S'$ inside at least one hypersphere of $s$ as well as the number of samples in $S$ inside at least one hypersphere of $s'$': $R(S, S') = \\sum\\limits_{s' \\in S'} f(s', S) + \\sum\\limits_{s \\in S} f(s, S')$.\n\n\\begin{comment}\nME$_\\text{Petersen}$, which returns a single-valued metric, can be defined as:\n\n\\begin{equation}\n    \\text{ME}_\\text{Petersen}(S,S') = 1 - \\text{max}\\Bigg(A\\Big(P,\\widehat{P}_\\text{Petersen}(S_r,S_e)\\Big),1\\Bigg).\n\\end{equation}\n\\end{comment}\n\nThe Petersen estimator was extended by \\newcite{schnabel} to incorporate multiple markings and recaptures. The population size estimate ($\\widehat{P}_\\text{Schnabel}$) is calculated from $T$ consecutive Petersen estimates:\n\n\\begin{equation}\n\\label{eq:schnabel}\n    \\widehat{P}_\\text{Schnabel}(S, S') = \\dfrac{C_T(S, S')M_T(S, S')}{R_T(S, S')},\n\\end{equation}\n\nThe set of marked samples at each iteration $t \\in \\{1,\\ldots,T\\}$, can be defined recursively as:\n\\begin{equation}\n    M(t, S, S') = \n    \\begin{cases}\n        S \\cup \\{s' \\in S'| f(s', S) = 1\\}, &\\text{if }t = 1,\\\\\n        S \\cup S', &\\text{if }t = T,\\\\\n        M(1,S,S') \\cup \\Bigg(\\bigcup\\limits_{i=1}^{t-1}\\text{NN}_k(s'_i, S')\\Bigg), &\\text{otherwise}.\n    \\end{cases}\n\\end{equation}\n\nME$_\\text{Schnabel}$'s first marking step is identical to ME$_\\text{Petersen}$'s single marking step ($M(1,S,S')=M(S,S')$), with all samples in $S$ as well as samples in $S'$ that are inside at least one hypersphere of $s$ being marked. By the final marking step, all samples will be marked since we iterated through all of them: $M_T(S,S') = |M(T,S,S')|$. For the other iterations, $1<t<T$, samples in $S'$ that are captured, \\textit{i.e.} are $k$-nearest neighbors of the $s'$ being iterated, but are not yet marked, are added to the marked set.\n\nAfter all recapture steps, which excludes the first marking step, the number of captured samples will be the number of samples in $S'$ and their respective $k$'th nearest neighbors as well as samples in $S$ that are inside the hypersphere of each $s'$: $C_T = (K+1)*|S'| + \\sum\\limits_{s' \\in S'} \\sum\\limits_{s \\in S} f(s, \\text{NN}_k(s', S'))$.\nSince all samples in $S$ have been marked in the first marking step, the number of total recaptures is the number of samples in $S$ inside the hypersphere of each $s'$ as well as the number of $k$-nearest neighbors of the iterated $s'$ that have already been marked: $R_T(S,S') = \\sum\\limits_{i = 1}^{|S'|} \\sum\\limits_{j = 1}^{|S|} \\Big(f(s_j, \\text{NN}_k(s'_i, S')) + |M(i,S,S') \\cap \\text{NN}_k(s'_i, S')|\\Big)$. \n\n\\begin{comment}\nME$_\\text{Schnabel}$ is thus defined as:\n\n\\begin{equation}\n    \\text{ME}_\\text{Schnabel}(S,S') = 1 - \\text{max}\\Bigg(A\\Big(P,\\widehat{P}_\\text{Schnabel}(S,S')\\Big), 1\\Bigg).\n\\end{equation}\n\\end{comment}\n\n\\begin{comment}\nAlternatively, \\cite{schumacher} proposed to use linear regression to estimate the number of samples marked prior to each time step and the proportion of samples marked in the current time step. At the last time step $T$, the population size can be estimated as follows:\n\n\n\\begin{equation}\n\\label{eq:schumacher}\n    \\widehat{P}_\\text{Schumacher}(S, S') = \\dfrac{C_T(S,S')M_T(S,S')^2}{R_T(S,S')M_T(S,S')},\n\\end{equation}\n\\end{comment}\n\n\\begin{comment}\nwith ME$_\\text{Schumacher}$ defined as:\n\n\\begin{equation}\n    \\text{ME}_\\text{Schumacher}(S,S') = 1 - \\text{max}\\Bigg(A\\Big(P,\\widehat{P}_\\text{Schumacher}(S,S')\\Big), 1\\Bigg).\n\\end{equation}\n\\end{comment}\n\nBoth ME$_\\text{Petersen}$ and ME$_\\text{Schnabel}$ are mark-recapture methods since they rely on marking and recapturing information to estimate the population size. We further used a maximum log-likelihood method: the model null from Program CAPTURE~\\cite{capture}. By considering the total number of marked samples ($M_T$) and the total number of captures ($C_{total}$) over $T$ iterations, with $T=|S \\cup S'|$, we iterate through several provisional population estimates ($P_\\text{CAPTURE} \\in \\mathbb{N}_{\\geq M}$) and compute their log-likelihood:\n\n\\begin{equation}\n\\label{eq:capture}\n\\begin{split}\n    L_n(P_\\text{CAPTURE};S,S') = \\ln\\Bigg(\\dfrac{P_\\text{CAPTURE}!}{(P_\\text{CAPTURE}-M_T(S,S'))!}\\Bigg) + C_{total}(S,S') \\times ln\\Big(C_{total}(S,S')\\Big)\\\\ + \\Big(TP_\\text{CAPTURE}-C_{total}(S,S')\\Big) \\times \\ln\\Big(TP_\\text{CAPTURE}-C_{total}(S,S')\\Big) - (TP_\\text{CAPTURE})\\ln(TP_\\text{CAPTURE}).\n\\end{split}\n\\end{equation}\n\nThe total number of captures corresponds to the number of samples in $S$ and $S'$ and their respective neighbors, as well as the number of samples in $S$ inside the hypersphere of a given $s'$ and vice-versa:\n$C_{total}(S,S') = \\sum\\limits_{s \\in S} \\sum\\limits_{s' \\in S'} ( f(s', \\text{NN}_k(s, S)) + |\\text{NN}_k(s, S)| ) + \\sum\\limits_{s' \\in S'} \\sum\\limits_{s \\in S} ( f(s, \\text{NN}_k(s', S')) + |\\text{NN}_k(s', S')|)$. \nThe final population estimate ($\\widehat{P}_\\text{CAPTURE}$) is then the estimate that maximizes Equation~\\ref{eq:capture}: \n\n\\begin{equation}\n\\label{eq:capture_2}\n\\widehat{P}_\\text{CAPTURE}(S,S') = \\argmax\\limits_{P_\\text{CAPTURE}}\\widehat{L_n}(P_\\text{CAPTURE};S,S').\n\\end{equation}\n\n\\begin{comment}\nME$_\\text{CAPTURE}$ is defined as:\n\n\\begin{equation}\n    \\text{ME}_\\text{CAPTURE}(S,S') = 1 - \\text{max}\\Bigg(A\\Big(P,\\widehat{P}_\\text{CAPTURE}(S,S')\\Big)\\Bigg).\n\\end{equation}\n\\end{comment}\n\nOur family of methods uses the accuracy loss of each estimator to compute their scores as follows:\n\n\\begin{equation}\n\\label{eq:mark_evaluate}\n    \\text{ME}_\\text{\\{Petersen, Schnabel, CAPTURE\\}}(S,S') = 1 - A\\Big(P,\\widehat{P}_\\text{\\{Petersen, Schnabel, CAPTURE\\}}(S,S')\\Big).\n\\end{equation}\n\nNote that, due to its iterative nature, ME$_\\text{Schnable}$ may be used to separately assess the quality and diversity of an evaluation set $S_e$ given a reference set $S_r$. More specifically, quality may be calculated by ME$_\\text{Schnable}(S_r, S_e)$, whereas diversity may be measured by ME$_\\text{Schnable}(S_e, S_r)$. On the other hand, ME$_\\text{Petersen}$ and ME$_\\text{CAPTURE}$ are single-valued metrics, since ME$_\\text{Petersen}(S_r,S_e)$ = ME$_\\text{Petersen}(S_e,S_r)$ and ME$_\\text{CAPTURE}(S_r,S_e)$ = ME$_\\text{CAPTURE}(S_e,S_r)$.\nWe refer to the Appendix for theoretical discussions. \n\nTo study the effects of different capture volumes, determined by different $K$, we used SBERT to get the sentence embeddings of 10k training sentences from MNLI~\\cite{williams2017broad} as the reference set, and 10k validation sentences as the evaluation set. Results are shown in Figure~\\ref{fig:num_k}, with $K \\in \\{1,\\ldots, 40\\}$. We observe that as $K$ increases, the population size estimated by all estimators converges to the true population size. In turn, the scores of our family of methods also converge to their maximum value of 1.\n\n\\begin{figure} \n\\caption{Effects of using different capture volumes, \\textit{i.e.} changing the number of neighbors ($K$), in the population size estimated of each estimator (left) and our respective method's score (right).}\n\\centering\n\\includegraphics[width=1.0\\textwidth]{Figures\/num_k.pdf}\n\\label{fig:num_k}\n\\end{figure}\n\n\n\\begin{comment}\n\\begin{equation}\n  \\text{ME}(S_r,S_e) =\n    \\begin{cases}\n        \\text{ME}_\\text{Petersen}(S_r,S_e) =  \\text{ME}_\\text{Petersen}(S_e,S_r) = 1 - A(\\widehat{P}_\\text{Petersen}(S_r,S_e)),&\\text{(single)}\\\\\n        \\text{ME}_\\text{CAPTURE}(S_r,S_e) = \\text{ME}_\\text{CAPTURE}(S_e,S_r) = 1 - A(\\widehat{P}_\\text{CAPTURE}(S_r,S_e)),&\\text{(single)}\\\\\n        \\text{ME}_\\text{Schnabel}(S_r,S_e) = 1 - A(\\widehat{P}_\\text{Schnabel}(S_r,S_e)),&\\text{(precision)}\\\\\n        \\text{ME}_\\text{Schnabel}(S_e,S_r) = 1 - A(\\widehat{P}_\\text{Schnabel}(S_e,S_r)),&\\text{(recall)}\\\\\n        \\text{ME}_\\text{Schumacher}(S_r,S_e) = 1 - A(\\widehat{P}_\\text{Schumacher}(S_r,S_e)),&\\text{(precision)}\\\\\n        \\text{ME}_\\text{Schumacher}(S_e,S_r) = 1 - A(\\widehat{P}_\\text{Schumacher}(S_e,S_r)).&\\text{(recall)}\\\\\n    \\end{cases}\n\\end{equation}\n\\end{comment}\n\n\\section{Synthetic experiments}\n\\label{sec:synthetic_experiments}\n\n\\begin{figure}\n\\caption{Mode collapse experiment where sentences from certain topics are dropped from the evaluation set. Quality (full lines) is expected to remain constant while diversity (dotted lines) is expected to drop as mode collapse aggravates.}\n\\centering\n\\includegraphics[width=1.0\\textwidth]{Figures\/mode_collapse.pdf}\n\\label{fig:mode_collapse}\n\\end{figure}\n\nTo simulate drops in quality and diversity, we used the MNLI dataset~\\cite{williams2017broad}, which consists of 433k sentence pairs annotated with one out of 5 possible topics. \nSince we were interested in the sentence-pair information for these experiments, we treated each sentence independently. Our reference set consists of sentences from the training set, whereas our evaluation set has sentences from the validation set. We kept the size of the reference and evaluation sets equal throughout our experiments to reduce possible method instabilities regarding sample size. We follow the experiments in \\newcite{semeniuta2018accurate} and simulate diversity loss by dropping sentences from certain topics (Section~\\ref{sec:mode_collapse}), whereas quality detriment is induced by swapping the words of each sentence (Section~\\ref{sec:swap}). For both experiments, we use SBERT embeddings from BERT-base pre-trained on SNLI\n~\\cite{snli} and MNLI~\\cite{williams2017broad} datasets ( \\textit{'bert-base-nli-mean-tokens'~\\footnote{https:\/\/github.com\/UKPLab\/sentence-transformers}}).\n\n\\subsection{Mode collapse}\n\\label{sec:mode_collapse}\n\nTo evaluate mode collapse, we dropped the sentences from specific topics from the evaluation set, containing sentences from all the available topics. Thus, the evaluation set only contains sentences from a subset of topics. What differs at each step is the number of topics included in the evaluation set: for example, dropping one topic means that the evaluation set only contains samples from the rest of the four available topics. The reference set remained unaltered throughout this process. We used 4k reference and 4k evaluation samples throughout this experiment.\n\nWe expect quality assessments to remain constant and diversity assessments to drop as fewer topics are represented in the evaluation set. For single-metric methods, we expect a detriment of the overall score throughout the mode dropping process. Results are presented in Figure~\\ref{fig:mode_collapse}, where we observe that our family of methods displays the expected behavior. IMPAR and FID also show expected performance (note that higher FID is worse since it represents a distance from the reference and evaluation distributions). On the other hand, PRD's quality assessment or precision drops significantly as mode collapse aggravates, which is not expected since the quality of the evaluation set is not affected in this experiment.\nWe further observe that all methods show high sensitivity when only one topic is represented in the evaluation set. For example, when 4 topics are dropped, IMPAR's quality assessment shifts by $\\approx 0.4$, while ME$_\\text{Schnabel}$ and ME$_\\text{Schnabel}$ shifts by $\\approx 0.5$. Hence, both methods show similar variance, despite the visualization contrast originated from different y-scales.\n\n\\subsection{Word swap}\n\\label{sec:swap}\n\nTo evaluate quality detriment, we swapped the words of each sentence in the evaluation set with a certain swap probability. Similarly to the mode collapse experiment, the reference set remains constant throughout this study. We used 10k reference and 10k evaluation samples. \n\\newcite{semeniuta2018accurate} showed that sentence embeddings derived from InferSent~\\cite{infersent} and Transformers~\\cite{vaswani2017attention} models were unable to detect similar quality perturbations. However, we observe that BERT embeddings can capture such quality detriment, observed by the variance of the scores of all the tested methods.\n\nFor this experiment, precision is expected to drop, while recall should remain constant. The overall score of single-valued metrics should deteriorate as the swap probability increases. Results are presented in Figure~\\ref{fig:swap}. Both the quality and diversity assessments of ME$_\\text{Schnabel}$ show the expected behavior, similarly to our single-metrics and FID. On the other hand, the recall or diversity assessment of both PRD and IMPAR drops unexpectedly. Moreover, IMPAR's quality or precision does not drop as significantly at higher swap probabilities, which is not desirable. \n\n\\begin{figure} \n\\caption{Swap experiment where words from each sentence in the evaluation set are swapped with a certain swap probability. Quality (full lines) is expected to drop while diversity (dotted lines) is expected to remain constant as the swap probability increases.}\n\\centering\n\\includegraphics[width=1.0\\textwidth]{Figures\/swap.pdf}\n\\label{fig:swap}\n\\end{figure}\n\n\\section{Language generation}\n\\label{sec:language_generation}\n\nWe further assessed the text generated by ten different language generation models presented in \\newcite{eval_all}. The models include a traditional language model and several types of autoencoders, namely variational, adversarial, adversarially regularized, and plain autoencoders. We used the human ratings assigned to each model's fluency presented in their work to study the correlation of our family of methods and other tested metrics to human evaluation. \nReverse and forward cross-entropy, \\textit{i.e.} Reverse CE and Forward CE, have been commonly used to assess text generation in the past~\\cite{eval_all,semeniuta2018accurate,lm_ppl}. \nThe reported Reverse CE and Forward CE results were taken from \\newcite{eval_all}, obtained by training a language model on English Gigaword~\\cite{napoles2012annotated}.\nWe refer to \\newcite{eval_all} for additional details.\n\n\\begin{comment}\n\\begin{table} \n\\vskip 0.15in\n\\begin{center}\n\\begin{small}\n\\begin{sc}\n\\centering\n\\begin{adjustbox}{width=1.0\\textwidth}\n\\begin{tabular}{ccccccccc}\n\\toprule\n{\\bf Corr.} & Reverse CE & {FID} & {PRD} & {IMPAR} & {\\bf ME$_\\text{Schnabel}$} & {\\bf ME$_\\text{Schumacher}$} & {\\bf ME$_\\text{Petersen}$} & {\\bf ME$_\\text{CAPTURE}$} \\\\\n\\midrule\n\\bf $r$ & 0.440 & 0.902 & 0.830 (K=5) & 0.745 (K=2) & \\bf 0.917 (K=1) & \\bf \\underline{0.919} (K=1) & 0.872 (K=1) & \\textbf{0.902 (K=1)}\\\\\n\\bf $k$ & 0.333 & 0.867 & 0.822 (K=15) & 0.778 (K=2) & \\bf \\underline{0.911} (K=1) & \\bf \\underline{0.911} (K=1) & \\bf \\underline{0.911} (K=1) & \\bf 0.867 (K=1)\\\\\n\\bf $p$ & 0.491 & 0.964 & 0.939 (K=15) & 0.903 (K=2)  & \\bf \\underline{0.976} (K=1) & \\bf \\underline{0.976} (K=1) & \\bf \\underline{0.976} (K=1) & \\bf 0.964 (K=1)\\\\\n\\bottomrule\n\\end{tabular}\n\\end{adjustbox}\n\\end{sc}\n\\end{small}\n\\end{center}\n\\caption{(best K, SBERT (Large)) Pearson, Kendall and Spearman correlations to system-level human evaluation of the different methods. Best values for each embedding type are underlined. Correlations where our methods outperform or match the other methods' performance are in bold. We observe that all contextual-based embeddings methods outperform the previous baselines on Absolute Pearson correlation. Moreover, contextual-based embeddings show higher correlation than non-contextual embeddings on Kendall. Absolute Pearson\/Kendall values for FID. Pearson scores for CAPTURE are with len(ref)+len(eval) occasions, whereas kendall and spearman scores are with 2 occasions.}\n\\label{tab:human_eval_correlations_best}\n\\end{table}\n\\end{comment}\n\n\\begin{table}[b] \n\\vskip 0.15in\n\\begin{center}\n\\begin{small}\n\\begin{sc}\n\\centering\n\\begin{tabular}{ccccccccc}\n\\toprule\n{\\bf Correlations} & Forward CE & Reverse CE & {FID} & {PRD} & {IMPAR} & {\\bf ME$_\\text{Schnabel}$} & {\\bf ME$_\\text{Petersen}$} & {\\bf ME$_\\text{CAPTURE}$} \\\\\n\\midrule\n\\bf Pearson $r$ & 0.606 & 0.440 & 0.902 & 0.830 & 0.745 & \\bf 0.917 & 0.872 & \\textbf{0.902}\\\\\n\\bf Kendall $k$ & 0.556 & 0.333 & 0.867 & 0.822 & 0.778 & \\bf \\underline{0.911} & \\bf \\underline{0.911} & \\bf 0.867\\\\\n\\bf Spearman $p$ & 0.697 & 0.491 & 0.964 & 0.939 & 0.903  & \\bf \\underline{0.976} & \\bf \\underline{0.976} & \\bf 0.964\\\\\n\\bottomrule\n\\end{tabular}\n\\end{sc}\n\\end{small}\n\\end{center}\n\\caption{Absolute correlations to human evaluation regarding the fluency of 10 different models with the best $K$. Best scores for each correlation are underlined. Bold values represent the correlations where our methods outperform or match all of the other methods' performance.\n}\n\\label{tab:human_eval_correlations_best}\n\\end{table}\n\nWe used SBERT embeddings from BERT-large trained on SNLI and MNLI datasets (\\textit{'bert-large-nli-mean-tokens'}) since they achieved the best-reported performance in\n~\\newcite{sbert}. Note that, since human evaluation is only related to each model's fluency, we only report the quality assessment scores for ME$_\\text{Schnabel}$, PRD, and IMPAR.\nFor our family of methods, as well as PRD and IMPAR, we iterate through $K$ values until correlation drops and present the results with the best $K$ of each method. Table~\\ref{tab:human_eval_correlations_best} shows the Pearson $p$, Kendall $k$, and Spearman $p$ correlations to human evaluation. \n\nOverall, our family of methods achieves the highest correlations to human evaluation. Note that despite being outperformed by FID, ME$_\\text{Petersen}$ still outperforms PRD and IMPAR across all correlations. Additional results with default $K$ for our family of methods, PRD, and IMPAR, as well as a comparison with InferSent and different SBERT embeddings, are provided in the Appendix.\n\n\\section{Contextualized word embeddings}\n\\label{sec:word_embeddings_mt}\n\nWe will now shift our focus to conditional language generation under finer-grained representations, \\textit{i.e.} contextualized word embeddings. We used embeddings from BERT-base fine-tuned on MNLI, identically to~\\newcite{moverscore}. For a fair comparison, we used the same embedding representations for all the methods in the following experiments. Due to the likely imbalance of reference and evaluation samples, we only report the quality assessment or precision of double-valued metrics. Similarly to Section\n~\\ref{sec:language_generation}, we report the results with the best $K$. Additional results with default $K$ can be found in the Appendix.\n\nUsing the information of the last layers of BERT has been shown to help in several downstream tasks~\\cite{liu_last_layers}. This has also been shown for language assessment, observed by, for example, the fact that the best performing layers of BERTScore are often latter layers~\\cite{bertscore}. MoverScore extends this thinking and aggregates the representations of the last five layers of BERT with $p$-means.\nFor our methods, instead of aggregating or routing this information, we use the vector representation from the last five layers for each specific word. Thus, each word has five representations, defined as five samples, in our scheme. See Figure~\\ref{fig:word_embeddings} for an illustration of this process. This also allows us to produce a better population estimate in the end due to the increase of the sample size.\n\n\\begin{figure} \n\\caption{When using contextualized word embeddings in our family of methods, each word is represented by five samples corresponding to the embeddings of the last five layers of BERT. We used the same procedure as Mark-Evaluate for PRD and IMPAR. On the other hand, MoverScore uses p-means to aggregate the information of the last five embeddings, and BERTScore uses the embeddings of a given layer for each word.}\n\\centering\n\\includegraphics[width=1.0\\textwidth]{Figures\/word_embeddings.pdf}\n\\label{fig:word_embeddings}\n\\end{figure}\n\n\\subsection{Machine translation}\n\\label{sec:machine_translation}\n\nWe start by assessing system-level machine translations from the WMT17 metrics task~\\cite{wmt17}. We evaluated the different methods on the five language pairs provided by \\newcite{moverscore}'s implementation \\footnote{https:\/\/github.com\/AIPHES\/emnlp19-moverscore}. Namely, we assess translations from Czech (cs), German (de), Russian (ru), Turkish (tr), and Chinese (zh) to English (en). Each language pair has around 3k reference with the respective evaluation translations from multiple systems (the number of systems for each language pair varies).\n\n\\begin{comment}\n\\begin{table*}\n  \\centering\n  \\begin{adjustbox}{width=1.0\\textwidth}\n  \\begin{tabular}{cccccccccccccccccc}\n    \\hline\n    \\textbf{Translations} & BERTScore & MoverScore & PRD & IMPAR & \\textbf{ME$_\\text{Schnabel}$} & \\textbf{ME$_\\text{Schumacher}$} & \\textbf{ME$_\\text{Petersen}$} & \\textbf{ME$_\\text{CAPTURE}$}\\\\\n    \\hline\n    \\bf cs-en ($r$) & 0.966 & 0.983 & \\underline{0.992} & 0.987 (K=2) & 0.989 (K=2) & 0.990 (K=2) & 0.988 (K=2) & 0.987 (K=2)\\\\ \\hline\n    \\bf de-en ($r$) & 0.859 & 0.920 & 0.769 & 0.934 (K=3) & \\bf 0.944 (K=2) & \\bf 0.946 (K=2) & \\bf \\underline{0.953} (K=1) & \\bf \\underline{0.953} (K=1)\\\\ \\hline\n    \\bf ru-en ($r$) & 0.868 & 0.921 & \\underline{0.933} & 0.896 (K=2) & 0.902 (K=2) & 0.900 (K=2) & 0.908 (K=2) & 0.908 (K=2)\\\\ \\hline\n    \\bf tr-en ($r$) & 0.938 & 0.931 & 0.935 & 0.959 (K=6) & \\bf \\underline{0.970} (K=6) & \\bf 0.969 (K=6) & \\bf 0.960 (K=2) & \\bf 0.959 (K=2)\\\\ \\hline\n    \\bf zh-en ($r$) & 0.894 & 0.943 & 0.889 & 0.933 (K=1) & \\bf 0.957 (K=1) & \\bf \\underline{0.959} (K=2) & 0.936 (K=1) & 0.936 (K=1)\\\\ \\hline\n    \\bf Average ($r$) & 0.905 & 0.940 & 0.904 & 0.942 & \\bf 0.952 & \\bf \\underline{0.953} & \\bf 0.949 & \\bf 0.949\\\\ \\hline\n  \\end{tabular}\n  \\end{adjustbox}\n  \\caption{System-level results for the WMT17 metrics task with best $K$. The best correlation of each language pair is underlined. Correlations where our methods outperform or match all of the other methods' are highlighted in bold.\n  }\n  \\label{tab:mt_word_best}\n\\end{table*}\n\\end{comment}\n\n\\begin{table*}\n  \\centering\n  \\begin{adjustbox}{width=1.0\\textwidth}\n  \\begin{tabular}{ccccccccccccccccc}\n    \\hline\n    \\textbf{Translations} & BERTScore & MoverScore & PRD & IMPAR & \\textbf{ME$_\\text{Schnabel}$}  & \\textbf{ME$_\\text{Petersen}$} & \\textbf{ME$_\\text{CAPTURE}$}\\\\\n    \\hline\n    \\bf cs-en ($r$) & 0.966 & 0.983 & \\underline{0.992} & 0.987 & 0.989 & 0.988 & 0.987\\\\ \\hline\n    \\bf de-en ($r$) & 0.859 & 0.920 & 0.769 & 0.934 & \\bf 0.944 & \\bf \\underline{0.953} & \\bf \\underline{0.953}\\\\ \\hline\n    \\bf ru-en ($r$) & 0.868 & 0.921 & \\underline{0.933} & 0.896 & 0.902 & 0.908 & 0.908\\\\ \\hline\n    \\bf tr-en ($r$) & 0.938 & 0.931 & 0.935 & 0.959 & \\bf \\underline{0.970} & \\bf 0.960 & \\bf 0.959\\\\ \\hline\n    \\bf zh-en ($r$) & 0.894 & 0.943 & 0.889 & 0.933 & \\bf \\underline{0.957} & 0.936 & 0.936 \\\\ \\hline\n    \\bf Average ($r$) & 0.905 & 0.940 & 0.904 & 0.942 & \\bf \\underline{0.952} & \\bf 0.949 & \\bf 0.949\\\\ \\hline\n  \\end{tabular}\n  \\end{adjustbox}\n  \\caption{Pearson correlations on the WMT17 metrics task. The best correlation of each language pair is underlined, while correlations of our methods that outperform or match any other method are in bold.\n  }\n  \\label{tab:mt_word_best}\n\\end{table*}\n\nPearson ($r$) correlations with human evaluation are presented in Table~\\ref{tab:mt_word_best}. Our family of metrics outperforms all the rest in several language pair translations. Moreover, our metrics show the highest correlation to human evaluation when considering the average correlation across all language pairs. BERTScore results were calculated using the embeddings from the last fifth layer of the aforementioned BERT model. \n\n\\subsection{Text summarization}\n\\label{sec:text_summarization}\n\nWe further assessed text summarization with the TAC-2009 dataset\\footnote{http:\/\/tac.nist.gov\/}, consisting of news articles from ten different topics, with four reference summaries and fifty-five evaluation summaries from summarization systems per article. We evaluate each evaluation summary independently, performing a summary-level evaluation. Two scores were assigned to each evaluation summary: the \\textit{pyramid score}, which evaluates the semantic similarity between the reference and evaluation summaries, and the \\textit{responsiveness score}, that measures the overall quality of the evaluation summary in terms of grammar and content.\n\n\nTable~\\ref{tab:human_eval_text_summarization} shows the Kendall ($k$), Pearson ($r$), and Spearman ($p$) correlation to human evaluation for each score. Considering Kendall and Spearman correlations, our family of methods outperforms all the rest on responsiveness score. Moreover, ME$_\\text{Petersen}$ and ME$_\\text{CAPTURE}$ outperform all methods on the above correlations on the pyramid score. \nConsidering Pearson correlation, our family of methods outperforms PRD, and at least one of our metrics consistently outperforms IMPAR on both scores. We hypothesize that the lower Pearson correlations of our metrics could be explained by the instability of the population estimation process due to the low amount of samples, \\textit{i.e. }reference and evaluation words.\n\n\\begin{comment}\n\\begin{table} \n\\vskip 0.15in\n\\begin{center}\n\\begin{small}\n\\begin{sc}\n\\centering\n\\begin{adjustbox}{width=1.0\\textwidth}\n\\begin{tabular}{cccccccccc}\n\\toprule\n\\bf Metrics & BERTScore & MoverScore & PRD & IMPAR & \\bf ME$_\\text{Schnabel}$ & \\bf ME$_\\text{Schumacher}$ & \\bf ME$_\\text{Petersen}$ & \\bf ME$_\\text{CAPTURE}$\\\\\n\\midrule\n\\bf Responsiveness ($k$) & - & 0.482 & 0.398 & 0.481 (K=5) & \\textbf{0.483} (K=4) & \\textbf{0.483} (K=4) & \\underline{\\textbf{0.487}} (K=4) & \\textbf{0.484} (K=5) \\\\\n\\bf Responsiveness ($r$) & 0.739 & \\underline{0.754} & 0.564 & 0.743 (K=5) & 0.739 (K=3) & 0.740 (K=3) & 0.683 (K=1) & 0.747 (K=4) \\\\\n\\bf Responsiveness ($p$) & 0.580 & 0.594 & 0.501 & 0.594 (K=5) & \\textbf{0.595} (K=4) & \\textbf{0.596} (K=4) & \\underline{\\textbf{0.598}} (K=5) & \\textbf{0.596} (K=7) \\\\\n\\midrule\n\\bf Pyramid ($k$) & - & 0.550 & 0.444 & 0.541 (K=5) & 0.548 (K=4) & \\textbf{0.550} (K=4) & \\textbf{0.555} (K=9) & \\underline{\\textbf{0.565}} (K=7)\\\\\n\\bf Pyramid ($r$) & 0.823 & \\underline{0.831} & 0.658 & 0.804 (K=7) & 0.813 (K=4) & 0.814 (K=4) & 0.770 (K=3) & 0.808 (K=4) \\\\\n\\bf Pyramid ($p$) & 0.703 & 0.701 & 0.588 & 0.693 (K=5) & 0.698 (K=4) & 0.700 (K=4) & \\textbf{0.704} (K=5) & \\underline{\\textbf{0.718}} (K=11)\\\\\n\\bottomrule\n\\end{tabular}\n\\end{adjustbox}\n\\end{sc}\n\\end{small}\n\\end{center}\n\\caption{Summary-level correlations to human evaluation on TAC 2009. Bold values represent correlations from our proposed metrics that outperform or math all the other methods. The best correlations for each score are underlined. BERTScore results were taken from~\\cite{moverscore}\n}\n\\label{tab:human_eval_text_summarization}\n\\end{table}\n\\end{comment}\n\n\\begin{table} \n\\vskip 0.15in\n\\begin{center}\n\\begin{small}\n\\begin{sc}\n\\centering\n\\begin{tabular}{ccccccccc}\n\\toprule\n\\bf Metrics & BERTScore & MoverScore & PRD & IMPAR & \\bf ME$_\\text{Schnabel}$ & \\bf ME$_\\text{Petersen}$ & \\bf ME$_\\text{CAPTURE}$\\\\\n\\midrule\n\\bf Responsiveness ($k$) & - & 0.482 & 0.398 & 0.481 & \\textbf{0.483} & \\underline{\\textbf{0.487}} & \\textbf{0.484} \\\\\n\\bf Responsiveness ($r$) & 0.739 & \\underline{0.754} & 0.564 & 0.743 & 0.739 & 0.683 & 0.747\\\\\n\\bf Responsiveness ($p$) & 0.580 & 0.594 & 0.501 & 0.594 & \\textbf{0.595} & \\underline{\\textbf{0.598}} & \\textbf{0.596} \\\\\n\\midrule\n\\bf Pyramid ($k$) & - & 0.550 & 0.444 & 0.541 & 0.548 & \\textbf{0.555} & \\underline{\\textbf{0.565}}\\\\\n\\bf Pyramid ($r$) & 0.823 & \\underline{0.831} & 0.658 & 0.804 & 0.813 & 0.770 & 0.808 \\\\\n\\bf Pyramid ($p$) & 0.703 & 0.701 & 0.588 & 0.693 & 0.698 & \\textbf{0.704} & \\underline{\\textbf{0.718}}\\\\\n\\bottomrule\n\\end{tabular}\n\\end{sc}\n\\end{small}\n\\end{center}\n\\caption{Summary-level correlations to human evaluation on TAC 2009. Bold values represent correlations from our proposed metrics that outperform or match all the other methods. The best correlations of each score are underlined. BERTScore results were taken from~\\newcite{moverscore}\n}\n\\label{tab:human_eval_text_summarization}\n\\end{table}\n\n\n\\section{Conclusion}\n\nIn this work, we present a family of methods derived from popular population size estimators that have been widely used in ecology in the past several decades. We show that our family of methods is able to assess language systems under different representations effectively, \\textit{i.e.} using contextualized word and sentence embeddings. Our methods show a high correlation to human evaluation on challenging language generation tasks as well as the desired sensitivity to detect mode collapse and quality detriment.\n\nIn the future, we would like to evaluate our family of metrics on image generation tasks, reinforcing the general applicability of our methods. Moreover, we plan to extend our family of methods to also cover popular open populations estimation methods, where the population size may vary over time. In the end, we hope that combining the information from closed and open population methods will improve the overall assessment of language systems, further fostering the adoption of ecology methods in NLP.\n\n\\bibliographystyle{coling}\n\n\\section{Credits}\n\nThis document has been adapted from the instructions for  \nCOLING-2018 proceedings compiled by Xiaodan Zhu and Zhiyuan Liu,\nwhich are, in turn, based on\nthe instructions for\nCOLING-2016 proceedings compiled by Hitoshi Isahara and Masao Utiyama,\nwhich are, in turn, based on\nthe instructions for\nCOLING-2014 proceedings compiled by Joachim Wagner, Liadh Kelly\nand Lorraine Goeuriot,\nwhich are, in turn, based on the instructions for earlier ACL proceedings,\nincluding \nthose for ACL-2014 by Alexander Koller and Yusuke Miyao,\nthose for ACL-2012 by Maggie Li and Michael\nWhite, those for ACL-2010 by Jing-Shing Chang and Philipp Koehn,\nthose for ACL-2008 by Johanna D. 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+{"text":"\n\\section*{Acknowledgement}\n\n\n\n\n\\section{Introduction}\\label{intro}\n\nFinite Element Analysis (FEA), which plays a critical role in modern engineering design and quality evaluations, provides numerical solutions for boundary value problems and simulates the distribution of physical fields with high precision \\cite{axelsson2001finite}. However, FEA generally requires high-performance computational resources and has an expensive time cost which becomes a bottleneck in the present manufacturing industry, especially for advanced time-sensitive techniques: inverse modeling, agile manufacturing, generative design system, etc. In order to solve this problem, many researchers have employed machine learning technologies to regress the distribution of physical fields to obtain instant simulation results while compromising the accuracy. Some researchers employed conventional machine learning methods (Support Vector Regression-SVG \\cite{capuano2019smart}, Artificial Neural Network - ANN \\cite{tamaddon2020data}) to approximate the physical field distribution and solve the boundary value problem. However, most methods are for specific boundary conditions and require extra training ahead of the application. This limits their capability of serving as a general FEA surrogate model, and their ability to solve abstract decision-making problems.\n\nRecently, with the rapid development of Deep Learning (DL) techniques, some researchers employed Convolutional Neural Network (CNN) as the backbone for solving physical field prediction problems, and found out that CNN based surrogate models cannot only solve basic boundary value problems (e.g., solid mechanics \\cite{nie2020stress, kantzos2019design, khadilkar2019deep}, fluid dynamics \\cite{kutz2017deep, guo2016convolutional, zhang2018application}, etc.) but also perform efficiently on some abstract design problems based on predicted physical field, for instance, topology optimization \\cite{lee2020cnn, nie2021topologygan,zhang2019deep,banga20183d}. However, CNN-based surrogate models still have inevitable disadvantages. Firstly, CNN can only accept input tensor with fixed input size, which becomes an obstacle for input tensor encoding, especially for some complex FEA conditions. Also, since CNN can only perform calculations on Euclidean coordinates, most CNN-based surrogate models employ square-shaped or cube-shaped meshes (2D quadrilateral or 3D hexahedron meshes with uniform side lengths) that require homogeneous grid generation on the input FEA model. The grid resolution sometimes cannot reach a sufficiently high level since the CNN usually cannot handle a large tensor input with hundreds of meshes along X\/Y\/Z axes due to the limitation of modern computation resources. Lastly, most FEA run on structured meshes (e.g., triangular meshes, or polygon meshes with different side lengths) that cannot directly fit into the CNN input tensor. Some researchers have presented alternate solutions to handle this problem, which is illustrated in Fig.~\\ref{fig:0_1_FemCompare}. We provide a von Mises stress distribution prediction for a 2D cantilever beam based on simulation results from ABAQUS \\cite{abaqus2011abaqus}. The beam is made of A36 steel with Young's modulus 200 GPa and Poisson's ratio of 0.32 and has an elliptical hole inside. Its left edge is fixed and an external force 1000N is exerted on the center of its right edge. The ground truth simulation result with triangular meshes (with an approximate global mesh size of 1.0 mm) is shown in Fig.~\\ref{fig:0_1_FemCompare}. One type of method for using CNN to regress the physical field is to approximate the original input model's geometry with a high-resolution voxelized representation, shown in \"Simulation results with quadrilateral meshes\" in Fig.~\\ref{fig:0_1_FemCompare} \\cite{nie2020stress}. However, the resolution is still limited by computational resources, and the simulation result can significantly change since the change in model shape can largely influence the global boundary condition, shown in Fig.~\\ref{fig:0_1_FemCompare}. The other type of method usually uses some post-process tasks, for instance, averaging the physical fields to fit into the square\/cube-shaped meshes. This will still lead to extra approximation error for the output results, shown in \"Average the results\" in Fig.~\\ref{fig:0_1_FemCompare}. All of those defects call for a more efficient DL surrogate model for FEA. The Graph Neural Network (GNN) comes into sight.\n\n\\begin{figure}[htp]\n\\vspace{-0.12in}\n\\centering\n\t\\includegraphics[keepaspectratio,width=0.7\\textwidth]{fig\/0_1_FemCompare.png}\n\t\\caption{Comparison between different FEM simulation and prediction results}\n\t\\label{fig:0_1_FemCompare}\n\t\\vspace{-0.12in}\n\\end{figure}\n\n\nThe GNN is a generalized version of CNN for non-Euclidean graphs, which has been widely applied to abstract decision-making and data-regression problems on different graph data structures, for instance: point cloud, triangular meshes, and molecular structures \\cite{alet2019graph, wang2020kalibre, belbute2020combining, guo2020semi, ogoke2020graph,  pfaff2020learning, sanchez2020learning, gilmer2017neural}. Since the structured meshes can be naturally embedded by an undirected graph $G$, many researchers investigated GNN-based surrogate models to solving FEA regression problems, show in Table~\\ref{tab:0_1_Compare}. It is assumed that an undirected graph $G=(V, E)$ represents the embedded structured meshes in an FEA model. Then, generally, $V={1, \\cdots, N}$ can be the set of N graph vertices (or graph nodes) denoting N geometrical node points in FEA while $ E \\subseteq V \\times V$ is the set of graph links (or graph edges) representing the connection of neighborhood geometrical nodes. Graph vertex $v$ contains features $f_v$ encoding the properties of a geometrical node point (e.g., coordinates, material properties, boundary conditions). Graph links generally employ adjacent matrix $E \\in [0,1]^{N \\times N}$ to describe the connection between geometrical node points. Nowadays, most applications employed GNN layers with Message Passing Neural Network (MPNN) framework \\cite{gilmer2017neural} whose updated message $m_v^{t+1}$ of graph vertex $v$ at layer $t+1$ can be represented by:\n\n\\begin{equation}\nm_v^{t+1} = \\sum_{\\omega \\in N(v)}^{} M_t (h_v^{t}, h_\\omega^{t}, e_{v\\omega})\n\\label{mpnn1}\n\\end{equation}\n\\begin{equation}\nh_v^{t+1}=U_t(h_v^{t}, m_v^{t+1})\n\\label{mpnn2}\n\\end{equation}\n\\begin{equation}\n\\hat{p}=R( \\left \\{ h_v^{T}|v \\in G \\right \\})\n\\label{mpnn3}\n\\end{equation} \n\\newline Where $M_t$, $U_t$ and $R$ are message function, vertex update function and readout function, respectively; $h$ represents the hidden state in GNN layers (e.g., $h_v^{t}$ is the hidden state of $v$ at layer $t$); $N(v)$ denotes the neighbors of $v$ in graph $G$; $e_{v\\omega}$ represents the graph link features; $\\hat{p}$ is the final prediction results. From the Eq.~(\\ref{mpnn2}), it can be seen that GNN layers can only pass the information of the embedded feature to its neighbor vertices. If the distance on a graph for passing an important message is relatively long, with current graph embedding approaches, more layers of GNN are needed. For instance, this can happen when GNN is employed to pass the information from the element with boundary condition information (external force or fixture, e.g., element marked by \"Element with force information\" in Fig.~\\ref{fig:0_1_FemCompare}) to the element whose stress field is required to be solved (\"Target element\" in Fig.~\\ref{fig:0_1_FemCompare}). One of the shortest message propagation paths from the boundary element to the target element is marked with red shown in Fig.~\\ref{fig:0_1_FemCompare}, which requires several layers of GNN to pass the information. However, deeply stacking the GNN layers can suffer from over-smoothing problems which cause indistinguishable graph-vertex embeddings and make GNN eventually not predict anything \\cite{li2020deepergcn, yang2020revisiting}. In general, this problem restricts most GNN models to around 3 layers with only short-range graph-vertex interactions and limited message passing distance.\n\n\\input{tab\/0_1_tab}\n\nGiven the limitations of GNN, from Table~\\ref{tab:0_1_Compare}, it can be seen that the GNN is effective on two types of physical field regression problems: The first type is the problems which input a small-scale graph data into GNN with generally no more than 100 graph vertices, and usually requires data pre-processing or model simplification \\cite{alet2019graph, wang2020kalibre}; the other type can solve the large-scale-graph problem, however, has only short-range graph-vertex interactions. This means the graph distance for the propagation of boundary information is relatively short since the elements embedding boundary information is relatively close to the target element whose physical field is required to be solved. For instance, Pfaff et al. \\cite{pfaff2020learning} developed an efficient GNN model for dynamic simulations based on triangular meshes. However, since the dataset contained sequential physical fields (displacement field, velocity field, stress field, etc.) and the input physical field for the current frame (or simulation step) was derived from the previous frame, the boundary condition information can slowly propagate from the boundary elements to the elements inside the model frame by frame. Then, the physical field of the target element was only determined by its surrounding elements that embed boundary information, which only requires the short-range graph-vertex interaction. The same happens when the regression model works on sequential physical field predictions(e.g., \\cite{sanchez2020learning}). Ogoke et al. \\cite{ogoke2020graph} predicted drag force on a large graph using velocity fields , but only elements around the streamlined object contributed to the total force. Although some researchers successfully developed efficient GNN based prediction model on large graphs, their applications were still limited. Belbute-Peres et al. \\cite{ belbute2020combining} developed a GNN based fluid dynamics prediction model on large graphs by interpolating PDE solver's results, which had a relatively short-range graph distance for the propagation of boundary-condition information since the result from the PDE solver carried the model's boundary condition. Guo et al. \\cite{ guo2020semi} developed a semi-supervised GNN model to predict optimized architected material structure on a large graph, but the selected training dataset was chosen from inside of the material which embedded boundary conditions. Also, the training needed to be conducted before each application in \\cite{ guo2020semi}.\n\nFrom above, it can be seen that there's no ready-to-use generalized machine learning approach for solving structured-mesh-based boundary value problems, especially when long-range graph-vertex interactions are involved. FEA generally requires structured meshes to accurately approximate the boundary of the input model geometry. It usually provides finer meshes to the model's delicate structures to improve the computation accuracy and employ coarse meshes to save the computational resource. The CNN-based FEA surrogate models have difficulty approximating the structured-mesh-based results, which becomes a bottleneck for the efficiency of related algorithms. The GNN-based approach shows its unique advantage of regressing structured-mesh-based results, however, a precise FEA model usually requires fine meshing to provide a high-resolution result, which can largely increase the graph size and therefore extends the propagation distance of the boundary information. This requires a deep GNN model and usually can cause the oversmoothing problem that is difficult to solve. Also, when GNN is applied to solve the static boundary value problems (different from the dynamic problems using the sequential physical field as its input with multiple frames\\cite{pfaff2020learning, sanchez2020learning}), the target element needs to gather all the boundary condition information in one frame (or simulation step) and then provides the prediction result. For instance, in the cantilever beam problem, the target element stress field is determined by all the elements with the boundary information (fixture, model's geometrical edges, external forces, shown in Fig.~\\ref{fig:0_1_FemCompare}), and this cannot be solved by any of the current approaches. Therefore, in order to solve these problems and provide a potential general DL-based surrogate regression model, we focus on the most fundamental solid mechanics problem - cantilever beam problem and develop a Boundary Oriented Graph Embedding (BOGE) approach. With the novel BOGE method, the GNN model works effectively on high-resolution triangular-mesh-based FEA simulation and predicts the accurate distribution of von Miss stress. The topological optimization predicted by GNN shows the capability of  our model to conduct abstract decision-making results based on the model's boundary condition. In this paper, Section \\ref{data} introduces our novel data representation which provides both multi-hop local adjacent information and shortcuts for global boundary information. Section \\ref{dataset} describes the details of our dataset preparation and the parametric designs for the FEA simulation results. Section \\ref{train} shows the model's training results that \\textcolor{blue}{achieves the regression with MSE of 0.011706 (2.41\\% MAPE) for stress field prediction and 0.002735 MSE (with 1.58\\% elements having error larger than 0.01) for topological optimization.} The outperformance of our model's prediction accuracy as well as its irreplaceable properties on regression of FEA results shows our models' un-substitutable significance on fast FEA simulations. We summarize the advantages of our GNN based surrogate model as follows:\n\\begin{itemize}[leftmargin=*]\n\\input{0_highlights}\n\\end{itemize}\n\n\n\n\\section{Data representation}\\label{data}\n\nThe physical field distribution is generally derived by resolving partial differential equations under the given boundary condition. For the stress analysis of the cantilever beam problem, denote the cantilever beam's geometry as $\\Omega$. Then, the global boundary condition $\\partial \\Omega$ of the problem can be expressed by:\n\n\\begin{equation}\n\\partial \\Omega = S_u + S_p\n\\label{bc}\n\\end{equation}\n\\newline Where $S_u$ and $S_p$ respectively represent the displacement (essential) and force\/stress (natural) boundary condition. In an FEA model, the algorithm usually employs discretized approach to decompose the input geometry $\\Omega$ into finite elements so that the complex boundary value problem can be simplified into solving the partial differential equation for each element. For the linear elasticity problem on isotropic materials, the canonical form of the relationship between mesh nodal displacement $\\mathbf{U}$ and the global boundary condition $\\mathbf{F}$ can be expressed by:\n\n\\begin{equation}\n\\mathbf{K}\\mathbf{U}=\\mathbf{F}\n\\label{hooke}\n\\end{equation}\n\\newline Where $\\mathbf{K}$ is the global stiffness matrix. Here, the global stiffness matrix $\\mathbf{K}$ is assembled by the elemental stiffness matrices $\\mathbf{k_e}$, which can be represented by an integration function of the element area $A$:\n\n\\begin{equation}\n\\mathbf{k_e} = \\int_{A}^{}\\mathbf{B_e^T}\\mathbf{C_e}\\mathbf{B_e} \\textup{d}A\n\\label{ke}\n\\end{equation}\n\\newline Where $\\mathbf{B_e}$ is the displacement differentiation matrix obtained by differentiation of displacements expressed through shape functions and nodal displacements; $\\mathbf{C_e}$ is the elasticity matrix constructed by Young's modulus ($E$) and Poisson's ratio ($\\nu$). Then, the nodal displacement field $\\mathbf{U}$ can be solved. Based on $\\mathbf{U}$, the stress field $\\mathbf{\\sigma} = [\\sigma_x, \\sigma_y, \\tau_{xy}, \\tau_{yx}]^T$ can be expressed by:\n\n\\begin{equation}\n\\mathbf{\\sigma} = \\mathbf{C_g}\\mathbf{B}\\mathbf{u}\n\\label{stress}\n\\end{equation}\n\\newline Where $\\mathbf{B}$ is the assembly of $\\mathbf{B_e}$ matrices; the block-diagonal matrix $\\mathbf{C_g}$ is constructed from elemental elasticity tensor $\\mathbf{C}$ that can be derived by $\\mathbf{C_e}$ \\cite{nie2020stress}. From the equation, it can be seen that the element nodal stress state and the nodal displacement are not only determined by the global boundary condition $\\partial \\Omega$ but also influenced by local boundaries determined by its neighbor elements due to the assembly of $\\mathbf{K}$. Specifically, if a GNN model is applied to solve the cantilever beam problem, the target element requires both long-range global boundary condition information and the short-range neighbor elements' properties to generate its stress field. Also, in a 2D cantilever-beam FEA simulation, three key items need to be fully defined to determine the final simulation solution: global boundary condition ($\\partial \\Omega$), predefined elements for the model geometry ($\\Omega$), and material properties. This section mainly introduces the BOGE approach which can comprehensively feed that information from the FEA model to the GNN.\n\n\\subsection{Graph vertex embedding method}\\label{chs}\n\nMost GNN-based FEA surrogate models embed mesh nodes as graph vertices for physical field simulation \\cite{belbute2020combining, guo2020semi, ogoke2020graph, pfaff2020learning, sanchez2020learning}. However, boundary conditions can be cast on mesh edges or mesh bodies, which is not convenient to be represented using element nodes. Also, it's natural to encode the material property inside the mesh body rather than mesh nodes, especially for the conditions when material property changes spatially, for instance, composite materials and non-homogeneous material. The mesh nodes are between the mesh bodies, which is hard to embed the change of mesh body properties. In order to solve this problem, according to the convention from the previous research \\cite{pfaff2020learning}, the mesh edge, as well as the mesh body and the mesh node, are considered as graph vertices. However, compared with the condition when only element nodes are considered, embedding each mesh node, mesh edge, and mesh body as separate graph vertices will expand the graph size 3 times for triangular-mesh-based simulations, which can run out of the computational resources or lead to an extremely long message passing distance for the propagation of boundary conditions. Therefore, we combine the properties of the mesh node, mesh edge, and mesh body into one graph vertex and embed the overall mesh features into the graph vertex to provide a compressed graph representation.\n\nThe format of our graph embedding features $f_v$ is illustrated in Fig.~\\ref{fig:1_1_Encoding}. The first several channels encode the center of the mesh which provides the approximate position information of the mesh body. Material properties follow those channels which provide mesh body properties. After that, we encode the geometrical properties of mesh edges using the edge's normal vector and its distance to the mesh center to fully define the edge's geometrical position. Each edge's geometrical property follows the edge's physical property that encodes the global boundary condition of the simulation model. Following the edge properties, additional point-based information can be added to the data representation, for instance, the external force cast on the mesh.\n\n\\begin{figure}[htp]\n\\vspace{-0.12in}\n\\centering\n\t\\includegraphics[keepaspectratio,width=0.7\\textwidth]{fig\/1_1_Encoding.png}\n\t\\caption{Encoding approach for triangular mesh}\n\t\\label{fig:1_1_Encoding}\n\t\\vspace{-0.12in}\n\\end{figure}\n\nIn Fig.~\\ref{fig:1_1_Encoding}, we provide an example of our data representation for solving the cantilever beam problem (details of the problem definition are explained in Sec.~\\ref{dataset}). For a 2D cantilever beam problem based on triangular-mesh simulation, the first two channels encode the mesh center $(x_e, y_e)$,  which is calculated by the average of the coordinates for all the mesh nodes. The following body property for a stress analysis simulation can be Young's modulus ($E$) and Poisson's ratio ($\\nu$) of the mesh material. After that, three mesh edges with their distances from the center ($d_k$) and their normal vectors ($(x_{\\vec{n}k}, y_{\\vec{n}k})$) are encoded (where k=1, 2, 3 representing each edge of the mesh). Following each edge geometry, the boundary condition is encoded by its boundary state ($S_{uk}$). We set the boundary state as 1 when the edge is on the outer contour of the input model geometry, while $S_{uk}=2$ for edges on fixtures. After that, the data representation encodes the external 2D force ($S_p$) by attaching its relative position ($(x_{F_{ext}}, y_{F_{ext}})$, calculated according to the mesh center) and its magnitude ($(F_{ext-x}, F_{ext-y})$) to the graph vertex features. Meshes without the external force need to be padded with zero to meet the requirement of graph embedding. Then, all the necessary information from a triangular mesh has been embedded into the graph vertex.\n\n\\subsection{Adjacency information}\\label{adj}\n\nThe GNN requires the input of graph links $E$ to figure out the message passing direction, which is encoded by the adjacency matrix. Conventionally, the mesh-based graph embedding is bounded by the geometry of the input mesh model  ($\\Omega$), which only considers the graph links between the mesh and its neighboring meshes. However, in order to shorten the message passing distance to use shallow layers of GNN to regress the physical fields, we provide shortcuts for both the global boundary condition and the local neighboring meshes.\n\nMost researchers focus on one-hop graph vertices, which are the meshes straightly connecting to the target mesh. Denote the distance between the target element and its neighbor element on the graph as $l_e$, then we represent the graph links between one-hop elements and the target element as $l_e=0$, shown in Fig.~\\ref{fig:1_2_Shortcut} with red graph links. However, in order to provide more local information to the target mesh and smooth the physical-field results, we also consider two-hop ($l_e=1$) and three-hop ($l_e=2$) graph links (marked with green and blue graph links respectively in Fig.~\\ref{fig:1_2_Shortcut}) and encode those to the adjacency matrix. Adding information about meshes' local connectivity assists to provide more details of local boundary conditions, whose performance is presented in Sec.~\\ref{train}.\n\n\\begin{figure}[htp]\n\\vspace{-0.12in}\n\\centering\n\t\\includegraphics[keepaspectratio,width=0.7\\textwidth]{fig\/1_2_Shortcut.png}\n\t\\caption{Encoding for graph connectivity and shortcuts}\n\t\\label{fig:1_2_Shortcut}\n\t\\vspace{-0.12in}\n\\end{figure}\n\nThen, we provide mesh connectivity between the target mesh and all the meshes that have the information of global boundary conditions ($\\partial \\Omega$) in order to provide the shortcut for solving the large-scale graph problem. In the cantilever beam problem, all the meshes with information of the model's geometrical edges, fixtures, and external forces, are connected to every internal mesh inside the beam, which is illustrated in Fig.~\\ref{fig:1_2_Shortcut}. With these graph linking shortcuts, the GNN serves for collecting weighted boundary conditions from the entire simulation model, other than only passing the mesh information to its neighborhood meshes with short-range distance. Therefore, the shallow-layer GNN structure can still regress high-resolution structured-mesh-based simulation results. The performance of this improvement is shown in detail in Sec.~\\ref{train}.\n\nIn this section, we present the BOGE approach to embed triangular-mesh-based FEA problems into the graph structure. This concept can be generalized to other polygon-mesh simulations with more mesh-edge information. Padding the channels with zeros to keep the uniform graph-vertex feature size for all the meshes, data representation can encode simulation with mixed type of structured mesh (e.g., triangular meshes and quadrilateral meshes) and will incur only a small amount of extra computational resources since GNN generally employs sparse tensor production during training and computing. Our BOGE approach largely compresses the graph size and provides shortcuts for the boundary conditions, which can improve the long-range graph-vertex interactions in boundary value problems. It is compatible with most MPNN-based GNN structures and can be generalized to solve other boundary value problems.\n\\section{Dataset preparation}\\label{dataset}\n\nIn order to validate the efficiency of this approach, we focus on two aspects of the cantilever beam problem: stress field prediction and topology optimization. We summarized all the cantilever beam models provided in \\cite{nie2020stress} and created a parameterized synthetic FEA simulation dataset. According to \\cite{zhang2018featurenet, peddireddy2021identifying}, we provide uniform random values to provide design parameters for cantilever beams to improve the repeatability and portability of the dataset. Details on dataset preparation are explained in this section.\n\n\\subsection{Stress field simulation}\\label{stress}\n\nAccording to \\cite{nie2020stress}, a balanced FEA simulation dataset with 9 types of cantilever beam models is summarized and shown in Fig.~\\ref{fig:3_1_BeamDataset} which includes all the beam geometries from \\cite{nie2020stress}. All the design parameters for the cantilever beams geometry are generated uniformly-randomly according to the range presented in Fig.~\\ref{fig:3_1_BeamDataset} with a minimum increment of 0.1mm. In the case of beam structures with holes, certain constraints have been considered to prevent thin walls with a thickness less than 1mm. The fixtures ($S_u=0$) in this dataset are always on the left edge of the cantilever beam while the external force ($S_p$) is exerted on a random position selected from the right edge of the beam. The magnitude and the angle of the force are uniform-randomly generated with a minimum increment of 100N and $\\frac{\\pi}{6}$ respectively, ranging from 100N to 1000N and from 0 to $2\\pi$. Although the generated cantilevered structures cannot represent all the beam structures, the selected ones represent the most common structures in the mechanical areas \\cite{nie2020stress}. Also, following the similar parametric design pattern, other features can be considered and added into the dataset in the future according to previous synthetic CAD design researches \\cite{zhang2018featurenet, peddireddy2021identifying}. \n\n\\begin{figure}[htp]\n\\vspace{-0.12in}\n\\centering\n\t\\includegraphics[keepaspectratio,width=0.7\\textwidth]{fig\/3_1_BeamDataset.png}\n\t\\caption{Dataset design parameters}\n\t\\label{fig:3_1_BeamDataset}\n\t\\vspace{-0.12in}\n\\end{figure}\n\nThe ground truth of the stress field distribution is calculated by a commercial FEA software - ABAQUS \\cite{abaqus2011abaqus}. The chosen material for the simulations is the A36 steel with a Young's modulus of 200 GPa and a Poisson's ratio of 0.32. The process of triangular tessellation for the cantilever beam structures is carried out by ABAQUS built-in functions with the approximate global mesh size of 1.0 mm. The settings for deviation factor and minimum size factor are both chosen to be the default value 0.1mm to guarantee a fine triangular-mesh approximation. ABAQUS conducts the FEA algorithm explained in Sec.~\\ref{data} to calculate the displacement field and the stress field for the cantilever beams. We choose von Mises stress ($\\sigma_{\\textup{vm}}$) as the target stress field for the final regression, which can be represented by:\n\n\\begin{equation}\n\\sigma_{\\textup{vm}} = \n\\sqrt{\\sigma^2_{x} + \\sigma^2_{y} -\n\\sigma_{x} \\sigma_{y} +\n3 \\, \\tau^2_{xy} }\n\\label{vm}\n\\end{equation}\n\nWe encode the triangular meshes with their boundary condition into the graph according to the BOGE approach explained in Sec.~\\ref{data}. The external force has been normalized in vertex features and the material properties have not been included since only one material is considered. Table~\\ref{tab:2_1_Dataset} provides details regarding the generated dataset.\n\n\\subsection{Topology optimization}\\label{topo}\n\nBased on the simulated stress field, we calculate topological optimization results to validate the abstract decision-making capability of our approach. ABAQUS employs density-based Simple Isotropic Material with Penalization (SIMP) method to optimize the cantilever beam structure. The optimization process works toward minimizing the compliance function $C(z)$ which is the sum of the strain energy of all the elements:\n\n\\begin{equation}\n\\left.\\begin{matrix}\nx_\\textup{min} & :C(z)=\\mathbf{U^{T}}\\mathbf{K}\\mathbf{U} & =\\sum_{e=1}^{N} (z_e)^{p} \\mathbf{u_e^{T}}\\mathbf{k_e}\\mathbf{u_e} \\\\ \n\\textup{s.t.} & \\frac{V(z)}{V_0}=V_f & \\\\ \n & \\mathbf{K}\\mathbf{U}=\\mathbf{F} & \\\\ \n & 0 < z_{min} & \n\\end{matrix}\\right\\}\n\\label{simp}\n\\end{equation}\n\\newline Where $V(z)$ and $V_0$ represent the optimized material volume and the model's design domain volume; $V_f$ is the volume friction, which has been chosen as 0.3 for the final optimization target. Variable $z$ represents the element material density that varies from $z_{min}$ (void if 0, but generally nonzero to avoid singularity) to 1 (the full solid); $p$ is the penalization power whose value is usually 3. \\cite{lee2020cnn, nie2021topologygan} Using ABAQUS built-in optimization algorithm, we set the maximum design cycle as 25 and solve the SIMP problem. The optimized cantilever beams have a final $V_f\\in (0.29,0.3]$ which satisfies the required design target. The optimized $z_e$ for each element has been exported as the regression ground truth for GNN to be trained.\n\nWe finish all the dataset preparation tasks on a personal computer with an Intel i7-10700k CPU. The data generation algorithm runs in less than 10 threads with the total CPU occupation less than 90\\%, which has not influenced the evaluation of the computation efficiency of ABAQUS. The average computation time for the stress prediction is around 11.5s while that for the topological optimization is around 489s. We provide a balanced dataset with 45k simulation results (5k simulations for each class from Table~\\ref{fig:3_1_BeamDataset} with 3.5k\/0.75k\/0.75k simulations for training\/validating\/testing subsets) for both stress field distribution predictions and topological optimizations. Another balanced dataset with 180k simulation results (20k simulations for each class from Table~\\ref{fig:3_1_BeamDataset} with 14k\/3k\/3k simulations for training\/validating\/testing subsets) for stress field predictions only is generated to investigate the influence of the dataset size. \n\n\\input{tab\/2_1_tab}\n\nThe parameterized synthetic dataset can be generalized to other fields related to the FEA surrogate DL model. The synthetic dataset sometimes has an expensive time cost that can become a bottleneck for our approach to be a general-purpose surrogate model. However, the dataset prepared by physical experiments can also serve the training task. The stress fields can be measured by installing strain gauges on real cantilever beams on the area whose position corresponds to that of the predefined meshes in the simulation. Multiple data can be generated within a short time when the external forces change. In this condition, the physical experiment can serve as a fast PDE solver which assists to generate sufficient data for the surrogate model. All of these ideas contribute to the realization of a generalized model with high-precision predictions under various working conditions.\n\n\n\\section{Neural Network Training and Results}\\label{train}\n\nIn order to present the efficiency of the BOGE approach, we compare its performance with the conventional graph embedding methods (without any graph shortcuts) on different state of art GNN structures (GCN \\cite{kipf2016semi}, GAT \\cite{velivckovic2017graph}, UNet \\cite{gao2019graph}, DeepGCN \\cite{li2019deepgcns, li2020deepergcn}). We choose the MSE (Mean Square Error) function as the loss function and the evaluation metric to quantify our prediction performance \\cite{nie2020stress, koeppe2020intelligent}. The predicted outputs \u2013 the magnitude of von Mises stress $\\sigma_\\textup{vm}$ and the volume density $z_e$, are all 1D tensors which can be written as $\\hat{p}=(\\hat{p}_1, \\hat{p}_1,..., \\hat{p}_n)$, while the ground truth results can be represented by $p=(p_1, p_2, ...,p_n)$. The MSE can be represented as:\n\n\\begin{equation}\n\\mathrm{MSE}=\\frac{1}{n}\\sum_{i=1}^{n}(p_i-\\hat{p}_i)^2\n\\label{mse}\n\\end{equation}\n\nAccording to \\cite{koeppe2020intelligent}, we also employ the MAPE (Mean Absolute Percentage Error) to present the error rate in percentage, which can be expressed by:\n\n\\begin{equation}\n\\mathrm{MAPE}=\\frac{100}{n}\\sum_{i=1}^{n}|\\frac{p_i-\\hat{p}_i}{p_i+\\varepsilon}|\n\\label{mape}\n\\end{equation}\n\\newline Where we set $\\varepsilon=0.01$ to avoid computation error when the ground truth result is zero or extremely small.\n\nAll the codes have been written in Pytorch (shared in Github\\footnote{\\url{https:\/\/github.com\/stxyfu\/Stress\\_GNN}}) with Pytorch Geometric library \\cite{fey2019fast} and run on an NVIDIA RTX 3090 GPU and an Intel i7-10700k CPU. For all training experiments, the ADAM optimization algorithm has been employed to adjust the hyperparameters in the model. The training batch size is set to 8 to make maximal use of GPU memory and provide a precise training gradient while the learning rate is 0.01 obtained through the grid search to provide the best performance of the final GNN models. 500 training epochs have been employed for each experiment to ensure the GNNs are all fully trained with suitable prediction accuracy. The entire training process for the selected GNN models (Table~\\ref{tab:3_3_stress} and Table~\\ref{tab:3_4_topo}) takes around 125 hours on 45k datasets and 493 hours on the 150k dataset. The training time can be largely reduced if multiple high-performance GPUs are employed. \n\n\\subsection{Stress field prediction with conventional graph embedding}\\label{deepGNN}\n\nWe first investigate the regression capability of GNNs with conventional graph embedding methods on the 45k stress field prediction dataset, shown in Table~\\ref{tab:2_1_Dataset}. Conventional graph embedding methods only consider one-hop graph links (with $l_e = 0$ in our condition) whose adjacency matrix does not contain any shortcuts for boundary elements. Using this graph embedding method, shallow layer GNN (\"GCN(3 layers)\" in Table~\\ref{tab:3_1_results}) cannot pass the boundary information to all the internal elements which cannot perform any regression at the end of the training. Fig.~\\ref{fig:ConvRes}(\\subref{fig:Conv3GCN}) illustrates that, with the 3-layer GCN, the boundary information only propagates through limited meshes near the model's geometrical boundary and cannot reach the internal elements inside the material. The deep-layer GNN can save the weighted features passed from other elements in its hidden state $h$ according to Eq.~(\\ref{mpnn1}) and provides a relatively long-range message passing. According to our statistics, the largest graph distance between two structured meshes is around $l_e=250$ which requires at least 8 layers of GNN for general MPNN layers. However, deep GNN usually incurs oversmoothing that makes the embedded features similar and prevents the further regression process. For example, in Fig.~\\ref{fig:ConvRes}(\\subref{fig:Conv8GCN}), the 8-layer GCN (\"GCN(8 layers)\" in Table~\\ref{tab:3_1_results}) cannot predict anything but only outputs uniform values after 500 training epochs, which illustrates this extreme oversmoothing phenomenon. In order to solve this problem, researchers have employed different methods to mitigate the influence of oversmoothing and realized a few state-of-art GNN structures with deep GNN layers. From those methods, the most popular anti-oversmoothing techniques are DeepGCN \\cite{li2019deepgcns, li2020deepergcn}, attention layers(e.g., GAT  \\cite{velivckovic2017graph}) and DropEdge \\cite{rong2019dropedge}. Some other GNN structures, though have been developed for other purposes, can also reach deep layers with sufficient accuracy, for instance, the g-U-net \\cite{gao2019graph}. However, most of these methods mainly work on classification problems. For the regression problem in our condition, extra validations are needed. Therefore, we have tested those models using the conventional graph embedding approach and shown those results in Table~\\ref{tab:3_1_results} and Fig.~\\ref{fig:ConvRes}.\n\n\n\\begin{table}[!h]\n\\centering\n\\includegraphics[keepaspectratio, width=0.7\\columnwidth]{fig\/3_2_Convention.png}\n\\input{tab\/3_1_tab}\n\\caption{Training results with different GNN structures and graph embedding methods (training\/validating\/testing accuracy are presented without regularization)}\n\\label{tab:3_1_results}\n\\end{table}\n\nDeepGCN employs the GEN (GENeralized Graph Convolution layers \\cite{li2020deepergcn}), layerNorm \\cite{li2019deepgcns} and ResNet structure \\cite{he2016deep}, making GNN sufficiently robust against oversmoothing and can reach 56 layers with high classification accuracy. However, 8-layers DeepGCN (\"DeepGCN(8 layers)\" in Table~\\ref{tab:3_1_results}) with the ResGCN backbone \\cite{li2019deepgcns} still cannot reach a high accuracy and performs some oversmoothing shown in Fig.~\\ref{fig:ConvRes}(\\subref{fig:Conv8DeepGCN}). GAT employs the attention mechanism that provides weighted attention coefficients to various graph features to ensure that GNN can pass the important information through deep layers. We employ the same DeepGCN backbone, but substitute GAT (with 4 attention heads) for GEN and try to improve the prediction accuracy, shown as \"GAT\" in Table~\\ref{tab:3_1_results}. According to our tests, the combination of GAT and the DeepGCN backbone performs better than simply stacking GAT layers on our dataset. However, the training result still appears to be incapable of obtaining acceptable results and shows an extremely oversmoothed result, shown in Fig.~\\ref{fig:ConvRes}(\\subref{fig:Conv8GAT}). The g-U-Net can encode and decode both high-level features and local features which can reach 5 layers in \\cite{gao2019graph}. However, each graph pooling layer drops both graph vertices (mesh elements) and graph links. When only one-hop graph links ($l_e=0$) are considered, with a very sparse adjacency matrix, stacking pooling layers in the g-U-net makes the adjacent matrix too sparse, so that only a few graph links remain in high-level feature scope, which cannot pass ample information to other elements through GNN and ruin the g-U-net's prediction capability. Also, a deeper g-U-net can still incur the extreme oversmoothing problem shown in Fig.~\\ref{fig:ConvRes}(\\subref{fig:ConvgUnet}) (with the training result shown as \"gUnet\" in Table~\\ref{tab:3_1_results}). Other anti-oversmoothing techniques like DropEdge can ruin the integrity of the graph, which is not suitable for our problem as we have tested. The overall training results show that deep-layer GNNs with the conventional graph embedding approach cannot simply regress the boundary value problem with long-range graph-vertex interactions.\n\n\n\n\\input{fig_results_conv\/fig_results_conv}\n\n\n\\subsection{Stress field prediction with BOGE}\\label{stress}\n\nThe BOGE approach shows its unique benefits for regressing the results of large-scale boundary value problems with simple shallow-layer GNNs. By providing shortcuts for both global boundary conditions and local information of neighboring elements, GNNs with BOGE bypass the message passing limitation of the MPNN framework, and all the internal elements can directly get the boundary condition information to solve the boundary value problem. We have employed the 3-layers DeepGCN and BOGE with $l_e=1,2,3$ graph links to validate its performance on the 45k stress field prediction dataset. The results are shown in Table~\\ref{tab:3_3_stress}. \n\nCompared with the predicted results from \"DeepGCN(3 layers)\" in Table~\\ref{tab:3_1_results}, BOGE with $l_e=0$ graph links (model \\#1 in Table~\\ref{tab:3_3_stress}) largely improves the prediction accuracy and reaches 0.016383 testing accuracy with MAPE of 2.85\\%, which is accurate enough to serve as an efficient FEA surrogate prediction model. The testing results has been compared for BOGE with different adjacency matrices that encodes local graph links with $l_e=1,2$. From Table~\\ref{tab:3_3_stress}, it can be seen that additional local mesh information cannot improve the prediction accuracy. The testing accuracy for $l_e=1$ (model \\#2 in Table~\\ref{tab:3_3_stress}) drops to 0.021737 with MAPE of 3.08\\% while that for $l_e=2$ (model \\#3 in Table~\\ref{tab:3_3_stress}) drops to 0.027302 with MAPE of 3.44\\%. More local graph edges provide worse prediction accuracy, which can be caused by message congestion \\cite{loukas2019graph} since more elements enlarge the size of the message passing at each training epoch. Also, the adjacency matrix with more graph links can lead to a denser tensor product which takes more computational resources. Therefore, we only employ BOGE with $l_e=0$ for the rest of the training task to provide a satisfying prediction result.\n\n\\begin{table}[!h]\n\\centering\n\\includegraphics[keepaspectratio, width=\\columnwidth]{fig\/3_3_StressGNN.png}\n\\caption{Stress prediction results (notations for neural network layers are explained in Table~\\ref{tab:3_1_results}; training\/validating\/testing accuracy are presented without regularization)}\n\\label{tab:3_3_stress}\n\\end{table}\n\nWe also consider the effect of the dataset size. Another training experiment with 180k simulation dataset (\"Stress prediction (180k)\" in Table~\\ref{tab:2_1_Dataset}) is generated by the same 3-layer DeepGCN structure (model \\#4 in Table~\\ref{tab:3_3_stress}). The testing accuracy increases to 0.011706 which shows a better training performance, but only slightly improves the MAPE that reaches 2.41\\%. This indicates that while the larger training dataset provides better results, a proper smaller dataset with around 45k data can still provide appropriate training results. Some of the predicted results from model \\#4 in Table~\\ref{tab:3_3_stress} are presented in Fig.~\\ref{fig:StressResults} which illustrates the effectiveness of the BOGE approach. The average prediction time for the GNN model (model \\#4 in Table~\\ref{tab:3_3_stress}) with GPU is 0.012ms which is far less than the ABAQUS computation time (around 11.5s in Table~\\ref{tab:2_1_Dataset}). Though the computation time for the GNN model does not include the time for data loading, input checking, and other unknown processes running in ABAQUS, the large difference of the order of magnitude validates the efficiency of our GNN surrogate model. The training loss and the validation\/testing accuracy for the stress predictions is shown in Fig.~\\ref{fig:loss}(\\subref{fig:StressLoss})\n\n\\begin{figure}[!htbp]\n\\centering\n\\begin{subfigure}{0.4\\textwidth}\n  \\centering\n  \\includegraphics[width=\\linewidth]{fig\/stressloss.png}\n  \\caption{Stress prediction}\n  \\label{fig:StressLoss}\n\\end{subfigure}%\n\\begin{subfigure}{0.4\\textwidth}\n  \\centering\n  \\includegraphics[width=\\linewidth]{fig\/topoloss.png}\n  \\caption{Topology optimization prediction}\n  \\label{fig:TopoLoss}\n\\end{subfigure}\n\\caption{Training loss, validation accuracy, and testing accuracy}\n\\label{fig:loss}\n\\end{figure}\n\n\\input{fig_stress\/fig_stress}\n\n\\subsection{Topology optimization prediction}\\label{stress}\n\nUsing the same method, we regress the topology optimization results to investigate the BOGE's potential on solving abstract decision-making problems. Using the same DeepGCN structure, we employ the same graph embedding inputs to regress the volume density $z_e$. Since topology optimization usually happens after the stress field simulation, we also consider the condition when the stress field information is added to the features of graph vertices. In this condition, the stress field information $\\mathbf{\\sigma} = [\\sigma_x, \\sigma_y, \\tau_{xy}]^T$ is attached to the graph-vertex features in BOGE approach. The training data employs BOGE with $l_e=1$ on 45k topology optimization dataset (\"Topology optimization (45k)\" in Table~\\ref{tab:2_1_Dataset}), and its training results are shown in Table~\\ref{tab:3_4_topo}. The dropout layers have been added after each layer with the dropout rate of 0.1 to provide a better prediction result.\n\n\\begin{table}[!h]\n\\centering\n\\includegraphics[keepaspectratio, width=\\columnwidth]{fig\/3_4_TopoGNN.png}\n\\caption{Topology optimization prediction results (notations for neural network layers are explained in Table~\\ref{tab:3_1_results}; training\/validating\/testing accuracy are presented without regularization)}\n\\label{tab:3_4_topo}\n\\end{table}\n\n\\begin{table}[!h]\n\\centering\n\\includegraphics[keepaspectratio, width=\\columnwidth]{fig\/3_5_TopoGNN.png}\n\\caption{Analyses for topology optimization prediction results (training\/validating\/testing accuracy are presented without regularization)}\n\\label{tab:3_5_acc}\n\\end{table}\n\nFrom the training results, it can be seen that BOGE without encoding the stress information (model \\#1 in Table~\\ref{tab:3_4_topo}) reaches 0.002735 testing accuracy, which is better than the best testing MSE - 0.059943 in \\cite{nie2021topologygan} with similar cantilever beam settings and the comparable number of meshes. However, the MAPE is relatively large and reaches 22.53\\% in testing accuracy. It is due to two effects. First, the range or the average value for the volume density in the topological optimization is relatively small. For stress prediction shown in Table~\\ref{tab:2_1_Dataset}, the average von Mises stress is around 2MPa while the maximum stress magnitude can reach 74.09MPa, which means the error divided by its ground truth value can be relatively small for the MAPE. Assume one element with the stress of 2MPa (which is the average stress) has the prediction error of 0.1MPa, the MAPE for this element is only 5\\%. However, from Table~\\ref{tab:2_1_Dataset}, the average volume density is around 0.29 whereas the maximum volume density is 1.0, which means that the same error can contribute largely to the final MAPE. For example, if the element with 0.29 volume density has the prediction error of 0.1, the MAPE for this element can be 34\\%. This explains one of the reasons that some researchers haven't employed the MAPE for evaluating topology optimization results \\cite{nie2021topologygan, zhang2019deep, banga20183d}. Further, the large MAPE is generated because some outliers around the edge of the optimized beam contribute to most of the errors. Assume that the resolution of the volume density is 0.01, which is also the $\\varepsilon$ in Eq.~(\\ref{mape}). From model \\#1 in Table~\\ref{tab:3_5_acc}, when only meshes with its ground-truth volume density larger than $\\varepsilon$ are considered (${p|p_i>0.01}$), the MAPE for those meshes is only 9.70\\%. Also, for each testing graph (or simulation model), around 22\\% - 23\\% of its meshes have the volume density larger than 0.01 for its ground truth value, but only 1.58\\% of the total meshes contribute to the errors larger than $\\varepsilon$. Some prediction results from model \\#1 in Table~\\ref{tab:3_4_topo} are shown in Fig.~\\ref{fig:TopoResults}. It can be seen that those 1.58\\% of meshes mostly are located primarily on the edges of the optimized beams. Since the boundary of the optimized beams usually has a volume density around $\\varepsilon=0.01$ as its ground truth value, any small prediction error can be largely amplified by the MAPE. However, regardless of the MAPE, the predicted volume density shown in Fig.~\\ref{fig:TopoResults} and its testing accuracy are sufficient to validate the BOGE's advantage in regressing complex decision-making problems. \n\nWith the stress field information, BOGE shows a worse prediction result, which reaches the MSE of 0.003378 for testing accuracy (model \\#2 in Table~\\ref{tab:3_4_topo}). Also, more outliers (6.00\\% from model \\#2 in Table~\\ref{tab:3_5_acc}) are found in this condition. Adding additional features to the graph vertex can lead to message congestion\\cite{loukas2019graph} and oversquashing\\cite{alon2020bottleneck}. Therefore, unlike the CNN-based surrogate model\\cite{nie2021topologygan}, GNN models are sensitive to the size of the input tensor and cannot be simply enhanced by adding extra input features. Balancing the input feature, hidden layer features, and the GNN layers can provide better results according to the GNN's property.\n\nThe topology optimization prediction shows BOGE's capability of making abstract decisions which benefits the smart design technology. The average prediction time for the topology-optimization GNN model (model \\#1 in Table~\\ref{tab:3_4_topo}) on GPU is around 0.011ms, similar to that of GNN stress field prediction, but far less than the ABAQUS computation time (around 489s in Table~\\ref{tab:2_1_Dataset}). This is because the SIMP method requires several iterations of computations that take much longer than the stress prediction. However, the GNN surrogate model can directly output the optimized result which largely saves the time cost. The overall performance of BOGE demonstrates its irreplaceable advantage in regressing the physical field on structured elements.The training loss and the validation\/testing accuracy for the stress predictions is shown in Fig.~\\ref{fig:loss}(\\subref{fig:TopoLoss})\n\n\\input{fig_topo\/fig_topo}\n\n\\section{Conclusion}\\label{conclusion}\n\nWe develop a Boundary Oriented Graph Embedding (BOGE) approach for the GNN-based FEA surrogate model, especially for regressing triangular-mesh-based FEA simulation results. The BOGE approach bypasses the limitations of the MPNN framework and provides shortcuts for both global boundary information and local multi-hop elements, which allows shallow-layer GNN to regress boundary value problems with long-range graph-vertex interactions. Working on solving the cantilever beam problem, the BOGE approach with 3-layer DeepGCN model \\textcolor{blue}{achieves the regression with MSE of 0.011706 (2.41\\% MAPE) for stress field prediction and 0.002735 MSE (with 1.58\\% elements having error larger than 0.01) for topological optimization.}, which validates the BOGE's efficiency in complex decision-making problems. The model has the potential to be applied to most physical-field fitting problems with any type of polygon meshes or mixed types of meshes. This study has potential limitations. The GNN performance is sensitive to the input-tensor shape and requires more theoretical analysis to investigate appropriate methods to improve the structure of GNN. Future work can focus on modifying GNN structures to obtain more accurate regression results and applying the BOGE approach to other boundary value problems.\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}