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transport/__init__.py

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+from .transport import ModelType, PathType, Sampler, Transport, WeightType
+
+
+def create_transport(
+    path_type="Linear",
+    prediction="velocity",
+    loss_weight=None,
+    train_eps=None,
+    sample_eps=None,
+    snr_type="uniform",
+    do_shift=True,
+    seq_len=1024,  # corresponding to 512x512
+):
+    """function for creating Transport object
+    **Note**: model prediction defaults to velocity
+    Args:
+    - path_type: type of path to use; default to linear
+    - learn_score: set model prediction to score
+    - learn_noise: set model prediction to noise
+    - velocity_weighted: weight loss by velocity weight
+    - likelihood_weighted: weight loss by likelihood weight
+    - train_eps: small epsilon for avoiding instability during training
+    - sample_eps: small epsilon for avoiding instability during sampling
+    """
+
+    if prediction == "noise":
+        model_type = ModelType.NOISE
+    elif prediction == "score":
+        model_type = ModelType.SCORE
+    else:
+        model_type = ModelType.VELOCITY
+
+    if loss_weight == "velocity":
+        loss_type = WeightType.VELOCITY
+    elif loss_weight == "likelihood":
+        loss_type = WeightType.LIKELIHOOD
+    else:
+        loss_type = WeightType.NONE
+
+    path_choice = {
+        "Linear": PathType.LINEAR,
+        "GVP": PathType.GVP,
+        "VP": PathType.VP,
+    }
+
+    path_type = path_choice[path_type]
+
+    if path_type in [PathType.VP]:
+        train_eps = 1e-5 if train_eps is None else train_eps
+        sample_eps = 1e-3 if train_eps is None else sample_eps
+    elif path_type in [PathType.GVP, PathType.LINEAR] and model_type != ModelType.VELOCITY:
+        train_eps = 1e-3 if train_eps is None else train_eps
+        sample_eps = 1e-3 if train_eps is None else sample_eps
+    else:  # velocity & [GVP, LINEAR] is stable everywhere
+        train_eps = 0
+        sample_eps = 0
+
+    # create flow state
+    state = Transport(
+        model_type=model_type,
+        path_type=path_type,
+        loss_type=loss_type,
+        train_eps=train_eps,
+        sample_eps=sample_eps,
+        snr_type=snr_type,
+        do_shift=do_shift,
+        seq_len=seq_len,
+    )
+
+    return state

transport/dpm_solver.py

@@ -0,0 +1,1386 @@
+# Copyright 2024 NVIDIA CORPORATION & AFFILIATES
+#
+# Licensed under the Apache License, Version 2.0 (the "License");
+# you may not use this file except in compliance with the License.
+# You may obtain a copy of the License at
+#
+#     http://www.apache.org/licenses/LICENSE-2.0
+#
+# Unless required by applicable law or agreed to in writing, software
+# distributed under the License is distributed on an "AS IS" BASIS,
+# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+# See the License for the specific language governing permissions and
+# limitations under the License.
+#
+# SPDX-License-Identifier: Apache-2.0
+
+# This file is modified from https://github.com/PixArt-alpha/PixArt-sigma
+import os
+
+import torch
+from tqdm import tqdm
+
+
+class NoiseScheduleFlow:
+    def __init__(
+        self,
+        schedule="discrete_flow",
+    ):
+        """Create a wrapper class for the forward SDE (EDM type)."""
+        self.T = 1
+        self.t0 = 0.001
+        self.schedule = schedule  # ['continuous', 'discrete_flow']
+        self.total_N = 1000
+
+    def marginal_log_mean_coeff(self, t):
+        """
+        Compute log(alpha_t) of a given continuous-time label t in [0, T].
+        """
+        return torch.log(self.marginal_alpha(t))
+
+    def marginal_alpha(self, t):
+        """
+        Compute alpha_t of a given continuous-time label t in [0, T].
+        """
+        return 1 - t
+
+    @staticmethod
+    def marginal_std(t):
+        """
+        Compute sigma_t of a given continuous-time label t in [0, T].
+        """
+        return t
+
+    def marginal_lambda(self, t):
+        """
+        Compute lambda_t = log(alpha_t) - log(sigma_t) of a given continuous-time label t in [0, T].
+        """
+        log_mean_coeff = self.marginal_log_mean_coeff(t)
+        log_std = torch.log(self.marginal_std(t))
+        return log_mean_coeff - log_std
+
+    @staticmethod
+    def inverse_lambda(lamb):
+        """
+        Compute the continuous-time label t in [0, T] of a given half-logSNR lambda_t.
+        """
+        return torch.exp(-lamb)
+
+
+def model_wrapper(
+    model,
+    noise_schedule,
+    model_type="noise",
+    model_kwargs={},
+    guidance_type="uncond",
+    condition=None,
+    unconditional_condition=None,
+    guidance_scale=1.0,
+    interval_guidance=[0, 1.0],
+    classifier_fn=None,
+    classifier_kwargs={},
+):
+    """Create a wrapper function for the noise prediction model.
+
+    DPM-Solver needs to solve the continuous-time diffusion ODEs. For DPMs trained on discrete-time labels, we need to
+    firstly wrap the model function to a noise prediction model that accepts the continuous time as the input.
+
+    We support four types of the diffusion model by setting `model_type`:
+
+        1. "noise": noise prediction model. (Trained by predicting noise).
+
+        2. "x_start": data prediction model. (Trained by predicting the data x_0 at time 0).
+
+        3. "v": velocity prediction model. (Trained by predicting the velocity).
+            The "v" prediction is derivation detailed in Appendix D of [1], and is used in Imagen-Video [2].
+
+            [1] Salimans, Tim, and Jonathan Ho. "Progressive distillation for fast sampling of diffusion models."
+                arXiv preprint arXiv:2202.00512 (2022).
+            [2] Ho, Jonathan, et al. "Imagen Video: High Definition Video Generation with Diffusion Models."
+                arXiv preprint arXiv:2210.02303 (2022).
+
+        4. "score": marginal score function. (Trained by denoising score matching).
+            Note that the score function and the noise prediction model follows a simple relationship:
+            ```
+                noise(x_t, t) = -sigma_t * score(x_t, t)
+            ```
+
+    We support three types of guided sampling by DPMs by setting `guidance_type`:
+        1. "uncond": unconditional sampling by DPMs.
+            The input `model` has the following format:
+            ``
+                model(x, t_input, **model_kwargs) -> noise | x_start | v | score
+            ``
+
+        2. "classifier": classifier guidance sampling [3] by DPMs and another classifier.
+            The input `model` has the following format:
+            ``
+                model(x, t_input, **model_kwargs) -> noise | x_start | v | score
+            ``
+
+            The input `classifier_fn` has the following format:
+            ``
+                classifier_fn(x, t_input, cond, **classifier_kwargs) -> logits(x, t_input, cond)
+            ``
+
+            [3] P. Dhariwal and A. Q. Nichol, "Diffusion models beat GANs on image synthesis,"
+                in Advances in Neural Information Processing Systems, vol. 34, 2021, pp. 8780-8794.
+
+        3. "classifier-free": classifier-free guidance sampling by conditional DPMs.
+            The input `model` has the following format:
+            ``
+                model(x, t_input, cond, **model_kwargs) -> noise | x_start | v | score
+            ``
+            And if cond == `unconditional_condition`, the model output is the unconditional DPM output.
+
+            [4] Ho, Jonathan, and Tim Salimans. "Classifier-free diffusion guidance."
+                arXiv preprint arXiv:2207.12598 (2022).
+
+
+    The `t_input` is the time label of the model, which may be discrete-time labels (i.e. 0 to 999)
+    or continuous-time labels (i.e. epsilon to T).
+
+    We wrap the model function to accept only `x` and `t_continuous` as inputs, and outputs the predicted noise:
+    ``
+        def model_fn(x, t_continuous) -> noise:
+            t_input = get_model_input_time(t_continuous)
+            return noise_pred(model, x, t_input, **model_kwargs)
+    ``
+    where `t_continuous` is the continuous time labels (i.e. epsilon to T). And we use `model_fn` for DPM-Solver.
+
+    ===============================================================
+
+    Args:
+        model: A diffusion model with the corresponding format described above.
+        noise_schedule: A noise schedule object, such as NoiseScheduleVP.
+        model_type: A `str`. The parameterization type of the diffusion model.
+                    "noise" or "x_start" or "v" or "score".
+        model_kwargs: A `dict`. A dict for the other inputs of the model function.
+        guidance_type: A `str`. The type of the guidance for sampling.
+                    "uncond" or "classifier" or "classifier-free".
+        condition: A pytorch tensor. The condition for the guided sampling.
+                    Only used for "classifier" or "classifier-free" guidance type.
+        unconditional_condition: A pytorch tensor. The condition for the unconditional sampling.
+                    Only used for "classifier-free" guidance type.
+        guidance_scale: A `float`. The scale for the guided sampling.
+        classifier_fn: A classifier function. Only used for the classifier guidance.
+        classifier_kwargs: A `dict`. A dict for the other inputs of the classifier function.
+    Returns:
+        A noise prediction model that accepts the noised data and the continuous time as the inputs.
+    """
+
+    def get_model_input_time(t_continuous):
+        """
+        Convert the continuous-time `t_continuous` (in [epsilon, T]) to the model input time.
+        For discrete-time DPMs, we convert `t_continuous` in [1 / N, 1] to `t_input` in [0, 1000 * (N - 1) / N].
+        For continuous-time DPMs, we just use `t_continuous`.
+        """
+        if noise_schedule.schedule == "discrete":
+            return (t_continuous - 1.0 / noise_schedule.total_N) * noise_schedule.total_N
+        elif noise_schedule.schedule == "discrete_flow":
+            return t_continuous * noise_schedule.total_N
+        else:
+            return t_continuous
+
+    def noise_pred_fn(x, t_continuous, cond=None):
+        t_input = get_model_input_time(t_continuous)
+        if cond is None:
+            output = model(x, t_input, **model_kwargs)
+        else:
+            output = model(x, t_input, cond, **model_kwargs)
+        if model_type == "noise":
+            return output
+        elif model_type == "x_start":
+            alpha_t, sigma_t = noise_schedule.marginal_alpha(t_continuous), noise_schedule.marginal_std(t_continuous)
+            return (x - expand_dims(alpha_t, x.dim()) * output) / expand_dims(sigma_t, x.dim())
+        elif model_type == "v":
+            alpha_t, sigma_t = noise_schedule.marginal_alpha(t_continuous), noise_schedule.marginal_std(t_continuous)
+            return expand_dims(alpha_t, x.dim()) * output + expand_dims(sigma_t, x.dim()) * x
+        elif model_type == "score":
+            sigma_t = noise_schedule.marginal_std(t_continuous)
+            return -expand_dims(sigma_t, x.dim()) * output
+        elif model_type == "flow":
+            _, sigma_t = noise_schedule.marginal_alpha(t_continuous), noise_schedule.marginal_std(t_continuous)
+            try:
+                noise = (1 - expand_dims(sigma_t, x.dim()).to(x)) * output + x
+            except:
+                noise = (1 - expand_dims(sigma_t, x.dim()).to(x)) * output[0] + x
+            return noise
+
+    def cond_grad_fn(x, t_input):
+        """
+        Compute the gradient of the classifier, i.e. nabla_{x} log p_t(cond | x_t).
+        """
+        with torch.enable_grad():
+            x_in = x.detach().requires_grad_(True)
+            log_prob = classifier_fn(x_in, t_input, condition, **classifier_kwargs)
+            return torch.autograd.grad(log_prob.sum(), x_in)[0]
+
+    def model_fn(x, t_continuous):
+        """
+        The noise predicition model function that is used for DPM-Solver.
+        """
+        guidance_tp = guidance_type
+        if guidance_tp == "uncond":
+            return noise_pred_fn(x, t_continuous)
+        elif guidance_tp == "classifier":
+            assert classifier_fn is not None
+            t_input = get_model_input_time(t_continuous)
+            cond_grad = cond_grad_fn(x, t_input)
+            sigma_t = noise_schedule.marginal_std(t_continuous)
+            noise = noise_pred_fn(x, t_continuous)
+            return noise - guidance_scale * expand_dims(sigma_t, x.dim()) * cond_grad
+        elif guidance_tp == "classifier-free":
+            if (
+                guidance_scale == 1.0
+                or unconditional_condition is None
+                or not (interval_guidance[0] < t_continuous[0] < interval_guidance[1])
+            ):
+                return noise_pred_fn(x, t_continuous, cond=condition)
+            else:
+                x_in = torch.cat([x] * 2)
+                t_in = torch.cat([t_continuous] * 2)
+                c_in = torch.cat([unconditional_condition, condition])
+                try:
+                    noise_uncond, noise = noise_pred_fn(x_in, t_in, cond=c_in).chunk(2)
+                except:
+                    noise_uncond, noise = noise_pred_fn(x_in, t_in, cond=c_in)[0].chunk(2)
+                return noise_uncond + guidance_scale * (noise - noise_uncond)
+
+    assert model_type in ["noise", "x_start", "v", "score", "flow"]
+    assert guidance_type in [
+        "uncond",
+        "classifier",
+        "classifier-free",
+    ]
+    return model_fn
+
+
+class DPM_Solver:
+    def __init__(
+        self,
+        model_fn,
+        noise_schedule,
+        algorithm_type="dpmsolver++",
+        correcting_x0_fn=None,
+        correcting_xt_fn=None,
+        thresholding_max_val=1.0,
+        dynamic_thresholding_ratio=0.995,
+    ):
+        """Construct a DPM-Solver.
+
+        We support both DPM-Solver (`algorithm_type="dpmsolver"`) and DPM-Solver++ (`algorithm_type="dpmsolver++"`).
+
+        We also support the "dynamic thresholding" method in Imagen[1]. For pixel-space diffusion models, you
+        can set both `algorithm_type="dpmsolver++"` and `correcting_x0_fn="dynamic_thresholding"` to use the
+        dynamic thresholding. The "dynamic thresholding" can greatly improve the sample quality for pixel-space
+        DPMs with large guidance scales. Note that the thresholding method is **unsuitable** for latent-space
+        DPMs (such as stable-diffusion).
+
+        To support advanced algorithms in image-to-image applications, we also support corrector functions for
+        both x0 and xt.
+
+        Args:
+            model_fn: A noise prediction model function which accepts the continuous-time input (t in [epsilon, T]):
+                ``
+                def model_fn(x, t_continuous):
+                    return noise
+                ``
+                The shape of `x` is `(batch_size, **shape)`, and the shape of `t_continuous` is `(batch_size,)`.
+            noise_schedule: A noise schedule object, such as NoiseScheduleVP.
+            algorithm_type: A `str`. Either "dpmsolver" or "dpmsolver++".
+            correcting_x0_fn: A `str` or a function with the following format:
+                ```
+                def correcting_x0_fn(x0, t):
+                    x0_new = ...
+                    return x0_new
+                ```
+                This function is to correct the outputs of the data prediction model at each sampling step. e.g.,
+                ```
+                x0_pred = data_pred_model(xt, t)
+                if correcting_x0_fn is not None:
+                    x0_pred = correcting_x0_fn(x0_pred, t)
+                xt_1 = update(x0_pred, xt, t)
+                ```
+                If `correcting_x0_fn="dynamic_thresholding"`, we use the dynamic thresholding proposed in Imagen[1].
+            correcting_xt_fn: A function with the following format:
+                ```
+                def correcting_xt_fn(xt, t, step):
+                    x_new = ...
+                    return x_new
+                ```
+                This function is to correct the intermediate samples xt at each sampling step. e.g.,
+                ```
+                xt = ...
+                xt = correcting_xt_fn(xt, t, step)
+                ```
+            thresholding_max_val: A `float`. The max value for thresholding.
+                Valid only when use `dpmsolver++` and `correcting_x0_fn="dynamic_thresholding"`.
+            dynamic_thresholding_ratio: A `float`. The ratio for dynamic thresholding (see Imagen[1] for details).
+                Valid only when use `dpmsolver++` and `correcting_x0_fn="dynamic_thresholding"`.
+
+        [1] Chitwan Saharia, William Chan, Saurabh Saxena, Lala Li, Jay Whang, Emily Denton, Seyed Kamyar Seyed Ghasemipour,
+            Burcu Karagol Ayan, S Sara Mahdavi, Rapha Gontijo Lopes, et al. Photorealistic text-to-image diffusion models
+            with deep language understanding. arXiv preprint arXiv:2205.11487, 2022b.
+        """
+        self.model = lambda x, t: model_fn(x, t.expand(x.shape[0]))
+        self.noise_schedule = noise_schedule
+        assert algorithm_type in ["dpmsolver", "dpmsolver++"]
+        self.algorithm_type = algorithm_type
+        if correcting_x0_fn == "dynamic_thresholding":
+            self.correcting_x0_fn = self.dynamic_thresholding_fn
+        else:
+            self.correcting_x0_fn = correcting_x0_fn
+        self.correcting_xt_fn = correcting_xt_fn
+        self.dynamic_thresholding_ratio = dynamic_thresholding_ratio
+        self.thresholding_max_val = thresholding_max_val
+        self.register_progress_bar()
+
+    def register_progress_bar(self, progress_fn=None):
+        """
+        Register a progress bar callback function
+
+        Args:
+            progress_fn: Callback function that takes current step and total steps as parameters
+        """
+        self.progress_fn = progress_fn if progress_fn is not None else lambda step, total: None
+
+    def update_progress(self, step, total_steps):
+        """
+        Update sampling progress
+
+        Args:
+            step: Current step number
+            total_steps: Total number of steps
+        """
+        if hasattr(self, "progress_fn"):
+            try:
+                self.progress_fn(step / total_steps, desc=f"Generating {step}/{total_steps}")
+            except:
+                self.progress_fn(step, total_steps)
+
+        else:
+            # If no progress_fn registered, use default empty function
+            pass
+
+    def dynamic_thresholding_fn(self, x0, t):
+        """
+        The dynamic thresholding method.
+        """
+        dims = x0.dim()
+        p = self.dynamic_thresholding_ratio
+        s = torch.quantile(torch.abs(x0).reshape((x0.shape[0], -1)), p, dim=1)
+        s = expand_dims(torch.maximum(s, self.thresholding_max_val * torch.ones_like(s).to(s.device)), dims)
+        x0 = torch.clamp(x0, -s, s) / s
+        return x0
+
+    def noise_prediction_fn(self, x, t):
+        """
+        Return the noise prediction model.
+        """
+        return self.model(x, t)
+
+    def data_prediction_fn(self, x, t):
+        """
+        Return the data prediction model (with corrector).
+        """
+        noise = self.noise_prediction_fn(x, t)
+        alpha_t, sigma_t = self.noise_schedule.marginal_alpha(t), self.noise_schedule.marginal_std(t)
+        x0 = (x - sigma_t * noise) / alpha_t
+        if self.correcting_x0_fn is not None:
+            x0 = self.correcting_x0_fn(x0, t)
+        return x0
+
+    def model_fn(self, x, t):
+        """
+        Convert the model to the noise prediction model or the data prediction model.
+        """
+        if self.algorithm_type == "dpmsolver++":
+            return self.data_prediction_fn(x, t)
+        else:
+            return self.noise_prediction_fn(x, t)
+
+    def get_time_steps(self, skip_type, t_T, t_0, N, device, shift=1.0):
+        """Compute the intermediate time steps for sampling.
+
+        Args:
+            skip_type: A `str`. The type for the spacing of the time steps. We support three types:
+                - 'logSNR': uniform logSNR for the time steps.
+                - 'time_uniform': uniform time for the time steps. (**Recommended for high-resolutional data**.)
+                - 'time_quadratic': quadratic time for the time steps. (Used in DDIM for low-resolutional data.)
+            t_T: A `float`. The starting time of the sampling (default is T).
+            t_0: A `float`. The ending time of the sampling (default is epsilon).
+            N: A `int`. The total number of the spacing of the time steps.
+            device: A torch device.
+        Returns:
+            A pytorch tensor of the time steps, with the shape (N + 1,).
+        """
+        if skip_type == "logSNR":
+            lambda_T = self.noise_schedule.marginal_lambda(torch.tensor(t_T).to(device))
+            lambda_0 = self.noise_schedule.marginal_lambda(torch.tensor(t_0).to(device))
+            logSNR_steps = torch.linspace(lambda_T.cpu().item(), lambda_0.cpu().item(), N + 1).to(device)
+            return self.noise_schedule.inverse_lambda(logSNR_steps)
+        elif skip_type == "time_uniform":
+            return torch.linspace(t_T, t_0, N + 1).to(device)
+        elif skip_type == "time_quadratic":
+            t_order = 2
+            t = torch.linspace(t_T ** (1.0 / t_order), t_0 ** (1.0 / t_order), N + 1).pow(t_order).to(device)
+            return t
+        elif skip_type == "time_uniform_flow":
+            betas = torch.linspace(t_T, t_0, N + 1).to(device)
+            sigmas = 1.0 - betas
+            sigmas = (shift * sigmas / (1 + (shift - 1) * sigmas)).flip(dims=[0])
+            return sigmas
+        else:
+            raise ValueError(
+                f"Unsupported skip_type {skip_type}, need to be 'logSNR' or 'time_uniform' or 'time_quadratic'"
+            )
+
+    def get_orders_and_timesteps_for_singlestep_solver(self, steps, order, skip_type, t_T, t_0, device):
+        """
+        Get the order of each step for sampling by the singlestep DPM-Solver.
+
+        We combine both DPM-Solver-1,2,3 to use all the function evaluations, which is named as "DPM-Solver-fast".
+        Given a fixed number of function evaluations by `steps`, the sampling procedure by DPM-Solver-fast is:
+            - If order == 1:
+                We take `steps` of DPM-Solver-1 (i.e. DDIM).
+            - If order == 2:
+                - Denote K = (steps // 2). We take K or (K + 1) intermediate time steps for sampling.
+                - If steps % 2 == 0, we use K steps of DPM-Solver-2.
+                - If steps % 2 == 1, we use K steps of DPM-Solver-2 and 1 step of DPM-Solver-1.
+            - If order == 3:
+                - Denote K = (steps // 3 + 1). We take K intermediate time steps for sampling.
+                - If steps % 3 == 0, we use (K - 2) steps of DPM-Solver-3, and 1 step of DPM-Solver-2 and 1 step of DPM-Solver-1.
+                - If steps % 3 == 1, we use (K - 1) steps of DPM-Solver-3 and 1 step of DPM-Solver-1.
+                - If steps % 3 == 2, we use (K - 1) steps of DPM-Solver-3 and 1 step of DPM-Solver-2.
+
+        ============================================
+        Args:
+            order: A `int`. The max order for the solver (2 or 3).
+            steps: A `int`. The total number of function evaluations (NFE).
+            skip_type: A `str`. The type for the spacing of the time steps. We support three types:
+                - 'logSNR': uniform logSNR for the time steps.
+                - 'time_uniform': uniform time for the time steps. (**Recommended for high-resolutional data**.)
+                - 'time_quadratic': quadratic time for the time steps. (Used in DDIM for low-resolutional data.)
+            t_T: A `float`. The starting time of the sampling (default is T).
+            t_0: A `float`. The ending time of the sampling (default is epsilon).
+            device: A torch device.
+        Returns:
+            orders: A list of the solver order of each step.
+        """
+        if order == 3:
+            K = steps // 3 + 1
+            if steps % 3 == 0:
+                orders = [3,] * (
+                    K - 2
+                ) + [2, 1]
+            elif steps % 3 == 1:
+                orders = [3,] * (
+                    K - 1
+                ) + [1]
+            else:
+                orders = [3,] * (
+                    K - 1
+                ) + [2]
+        elif order == 2:
+            if steps % 2 == 0:
+                K = steps // 2
+                orders = [
+                    2,
+                ] * K
+            else:
+                K = steps // 2 + 1
+                orders = [2,] * (
+                    K - 1
+                ) + [1]
+        elif order == 1:
+            K = 1
+            orders = [
+                1,
+            ] * steps
+        else:
+            raise ValueError("'order' must be '1' or '2' or '3'.")
+        if skip_type == "logSNR":
+            # To reproduce the results in DPM-Solver paper
+            timesteps_outer = self.get_time_steps(skip_type, t_T, t_0, K, device)
+        else:
+            timesteps_outer = self.get_time_steps(skip_type, t_T, t_0, steps, device)[
+                torch.cumsum(
+                    torch.tensor(
+                        [
+                            0,
+                        ]
+                        + orders
+                    ),
+                    0,
+                ).to(device)
+            ]
+        return timesteps_outer, orders
+
+    def denoise_to_zero_fn(self, x, s):
+        """
+        Denoise at the final step, which is equivalent to solve the ODE from lambda_s to infty by first-order discretization.
+        """
+        return self.data_prediction_fn(x, s)
+
+    def dpm_solver_first_update(self, x, s, t, model_s=None, return_intermediate=False):
+        """
+        DPM-Solver-1 (equivalent to DDIM) from time `s` to time `t`.
+
+        Args:
+            x: A pytorch tensor. The initial value at time `s`.
+            s: A pytorch tensor. The starting time, with the shape (1,).
+            t: A pytorch tensor. The ending time, with the shape (1,).
+            model_s: A pytorch tensor. The model function evaluated at time `s`.
+                If `model_s` is None, we evaluate the model by `x` and `s`; otherwise we directly use it.
+            return_intermediate: A `bool`. If true, also return the model value at time `s`.
+        Returns:
+            x_t: A pytorch tensor. The approximated solution at time `t`.
+        """
+        ns = self.noise_schedule
+        dims = x.dim()
+        lambda_s, lambda_t = ns.marginal_lambda(s), ns.marginal_lambda(t)
+        h = lambda_t - lambda_s
+        log_alpha_s, log_alpha_t = ns.marginal_log_mean_coeff(s), ns.marginal_log_mean_coeff(t)
+        sigma_s, sigma_t = ns.marginal_std(s), ns.marginal_std(t)
+        alpha_t = torch.exp(log_alpha_t)
+
+        if self.algorithm_type == "dpmsolver++":
+            phi_1 = torch.expm1(-h)
+            if model_s is None:
+                model_s = self.model_fn(x, s)
+            x_t = sigma_t / sigma_s * x - alpha_t * phi_1 * model_s
+            if return_intermediate:
+                return x_t, {"model_s": model_s}
+            else:
+                return x_t
+        else:
+            phi_1 = torch.expm1(h)
+            if model_s is None:
+                model_s = self.model_fn(x, s)
+            x_t = torch.exp(log_alpha_t - log_alpha_s) * x - (sigma_t * phi_1) * model_s
+            if return_intermediate:
+                return x_t, {"model_s": model_s}
+            else:
+                return x_t
+
+    def singlestep_dpm_solver_second_update(
+        self, x, s, t, r1=0.5, model_s=None, return_intermediate=False, solver_type="dpmsolver"
+    ):
+        """
+        Singlestep solver DPM-Solver-2 from time `s` to time `t`.
+
+        Args:
+            x: A pytorch tensor. The initial value at time `s`.
+            s: A pytorch tensor. The starting time, with the shape (1,).
+            t: A pytorch tensor. The ending time, with the shape (1,).
+            r1: A `float`. The hyperparameter of the second-order solver.
+            model_s: A pytorch tensor. The model function evaluated at time `s`.
+                If `model_s` is None, we evaluate the model by `x` and `s`; otherwise we directly use it.
+            return_intermediate: A `bool`. If true, also return the model value at time `s` and `s1` (the intermediate time).
+            solver_type: either 'dpmsolver' or 'taylor'. The type for the high-order solvers.
+                The type slightly impacts the performance. We recommend to use 'dpmsolver' type.
+        Returns:
+            x_t: A pytorch tensor. The approximated solution at time `t`.
+        """
+        if solver_type not in ["dpmsolver", "taylor"]:
+            raise ValueError(f"'solver_type' must be either 'dpmsolver' or 'taylor', got {solver_type}")
+        if r1 is None:
+            r1 = 0.5
+        ns = self.noise_schedule
+        lambda_s, lambda_t = ns.marginal_lambda(s), ns.marginal_lambda(t)
+        h = lambda_t - lambda_s
+        lambda_s1 = lambda_s + r1 * h
+        s1 = ns.inverse_lambda(lambda_s1)
+        log_alpha_s, log_alpha_s1, log_alpha_t = (
+            ns.marginal_log_mean_coeff(s),
+            ns.marginal_log_mean_coeff(s1),
+            ns.marginal_log_mean_coeff(t),
+        )
+        sigma_s, sigma_s1, sigma_t = ns.marginal_std(s), ns.marginal_std(s1), ns.marginal_std(t)
+        alpha_s1, alpha_t = torch.exp(log_alpha_s1), torch.exp(log_alpha_t)
+
+        if self.algorithm_type == "dpmsolver++":
+            phi_11 = torch.expm1(-r1 * h)
+            phi_1 = torch.expm1(-h)
+
+            if model_s is None:
+                model_s = self.model_fn(x, s)
+            x_s1 = (sigma_s1 / sigma_s) * x - (alpha_s1 * phi_11) * model_s
+            model_s1 = self.model_fn(x_s1, s1)
+            if solver_type == "dpmsolver":
+                x_t = (
+                    (sigma_t / sigma_s) * x
+                    - (alpha_t * phi_1) * model_s
+                    - (0.5 / r1) * (alpha_t * phi_1) * (model_s1 - model_s)
+                )
+            elif solver_type == "taylor":
+                x_t = (
+                    (sigma_t / sigma_s) * x
+                    - (alpha_t * phi_1) * model_s
+                    + (1.0 / r1) * (alpha_t * (phi_1 / h + 1.0)) * (model_s1 - model_s)
+                )
+        else:
+            phi_11 = torch.expm1(r1 * h)
+            phi_1 = torch.expm1(h)
+
+            if model_s is None:
+                model_s = self.model_fn(x, s)
+            x_s1 = torch.exp(log_alpha_s1 - log_alpha_s) * x - (sigma_s1 * phi_11) * model_s
+            model_s1 = self.model_fn(x_s1, s1)
+            if solver_type == "dpmsolver":
+                x_t = (
+                    torch.exp(log_alpha_t - log_alpha_s) * x
+                    - (sigma_t * phi_1) * model_s
+                    - (0.5 / r1) * (sigma_t * phi_1) * (model_s1 - model_s)
+                )
+            elif solver_type == "taylor":
+                x_t = (
+                    torch.exp(log_alpha_t - log_alpha_s) * x
+                    - (sigma_t * phi_1) * model_s
+                    - (1.0 / r1) * (sigma_t * (phi_1 / h - 1.0)) * (model_s1 - model_s)
+                )
+        if return_intermediate:
+            return x_t, {"model_s": model_s, "model_s1": model_s1}
+        else:
+            return x_t
+
+    def singlestep_dpm_solver_third_update(
+        self,
+        x,
+        s,
+        t,
+        r1=1.0 / 3.0,
+        r2=2.0 / 3.0,
+        model_s=None,
+        model_s1=None,
+        return_intermediate=False,
+        solver_type="dpmsolver",
+    ):
+        """
+        Singlestep solver DPM-Solver-3 from time `s` to time `t`.
+
+        Args:
+            x: A pytorch tensor. The initial value at time `s`.
+            s: A pytorch tensor. The starting time, with the shape (1,).
+            t: A pytorch tensor. The ending time, with the shape (1,).
+            r1: A `float`. The hyperparameter of the third-order solver.
+            r2: A `float`. The hyperparameter of the third-order solver.
+            model_s: A pytorch tensor. The model function evaluated at time `s`.
+                If `model_s` is None, we evaluate the model by `x` and `s`; otherwise we directly use it.
+            model_s1: A pytorch tensor. The model function evaluated at time `s1` (the intermediate time given by `r1`).
+                If `model_s1` is None, we evaluate the model at `s1`; otherwise we directly use it.
+            return_intermediate: A `bool`. If true, also return the model value at time `s`, `s1` and `s2` (the intermediate times).
+            solver_type: either 'dpmsolver' or 'taylor'. The type for the high-order solvers.
+                The type slightly impacts the performance. We recommend to use 'dpmsolver' type.
+        Returns:
+            x_t: A pytorch tensor. The approximated solution at time `t`.
+        """
+        if solver_type not in ["dpmsolver", "taylor"]:
+            raise ValueError(f"'solver_type' must be either 'dpmsolver' or 'taylor', got {solver_type}")
+        if r1 is None:
+            r1 = 1.0 / 3.0
+        if r2 is None:
+            r2 = 2.0 / 3.0
+        ns = self.noise_schedule
+        lambda_s, lambda_t = ns.marginal_lambda(s), ns.marginal_lambda(t)
+        h = lambda_t - lambda_s
+        lambda_s1 = lambda_s + r1 * h
+        lambda_s2 = lambda_s + r2 * h
+        s1 = ns.inverse_lambda(lambda_s1)
+        s2 = ns.inverse_lambda(lambda_s2)
+        log_alpha_s, log_alpha_s1, log_alpha_s2, log_alpha_t = (
+            ns.marginal_log_mean_coeff(s),
+            ns.marginal_log_mean_coeff(s1),
+            ns.marginal_log_mean_coeff(s2),
+            ns.marginal_log_mean_coeff(t),
+        )
+        sigma_s, sigma_s1, sigma_s2, sigma_t = (
+            ns.marginal_std(s),
+            ns.marginal_std(s1),
+            ns.marginal_std(s2),
+            ns.marginal_std(t),
+        )
+        alpha_s1, alpha_s2, alpha_t = torch.exp(log_alpha_s1), torch.exp(log_alpha_s2), torch.exp(log_alpha_t)
+
+        if self.algorithm_type == "dpmsolver++":
+            phi_11 = torch.expm1(-r1 * h)
+            phi_12 = torch.expm1(-r2 * h)
+            phi_1 = torch.expm1(-h)
+            phi_22 = torch.expm1(-r2 * h) / (r2 * h) + 1.0
+            phi_2 = phi_1 / h + 1.0
+            phi_3 = phi_2 / h - 0.5
+
+            if model_s is None:
+                model_s = self.model_fn(x, s)
+            if model_s1 is None:
+                x_s1 = (sigma_s1 / sigma_s) * x - (alpha_s1 * phi_11) * model_s
+                model_s1 = self.model_fn(x_s1, s1)
+            x_s2 = (
+                (sigma_s2 / sigma_s) * x
+                - (alpha_s2 * phi_12) * model_s
+                + r2 / r1 * (alpha_s2 * phi_22) * (model_s1 - model_s)
+            )
+            model_s2 = self.model_fn(x_s2, s2)
+            if solver_type == "dpmsolver":
+                x_t = (
+                    (sigma_t / sigma_s) * x
+                    - (alpha_t * phi_1) * model_s
+                    + (1.0 / r2) * (alpha_t * phi_2) * (model_s2 - model_s)
+                )
+            elif solver_type == "taylor":
+                D1_0 = (1.0 / r1) * (model_s1 - model_s)
+                D1_1 = (1.0 / r2) * (model_s2 - model_s)
+                D1 = (r2 * D1_0 - r1 * D1_1) / (r2 - r1)
+                D2 = 2.0 * (D1_1 - D1_0) / (r2 - r1)
+                x_t = (
+                    (sigma_t / sigma_s) * x
+                    - (alpha_t * phi_1) * model_s
+                    + (alpha_t * phi_2) * D1
+                    - (alpha_t * phi_3) * D2
+                )
+        else:
+            phi_11 = torch.expm1(r1 * h)
+            phi_12 = torch.expm1(r2 * h)
+            phi_1 = torch.expm1(h)
+            phi_22 = torch.expm1(r2 * h) / (r2 * h) - 1.0
+            phi_2 = phi_1 / h - 1.0
+            phi_3 = phi_2 / h - 0.5
+
+            if model_s is None:
+                model_s = self.model_fn(x, s)
+            if model_s1 is None:
+                x_s1 = (torch.exp(log_alpha_s1 - log_alpha_s)) * x - (sigma_s1 * phi_11) * model_s
+                model_s1 = self.model_fn(x_s1, s1)
+            x_s2 = (
+                (torch.exp(log_alpha_s2 - log_alpha_s)) * x
+                - (sigma_s2 * phi_12) * model_s
+                - r2 / r1 * (sigma_s2 * phi_22) * (model_s1 - model_s)
+            )
+            model_s2 = self.model_fn(x_s2, s2)
+            if solver_type == "dpmsolver":
+                x_t = (
+                    (torch.exp(log_alpha_t - log_alpha_s)) * x
+                    - (sigma_t * phi_1) * model_s
+                    - (1.0 / r2) * (sigma_t * phi_2) * (model_s2 - model_s)
+                )
+            elif solver_type == "taylor":
+                D1_0 = (1.0 / r1) * (model_s1 - model_s)
+                D1_1 = (1.0 / r2) * (model_s2 - model_s)
+                D1 = (r2 * D1_0 - r1 * D1_1) / (r2 - r1)
+                D2 = 2.0 * (D1_1 - D1_0) / (r2 - r1)
+                x_t = (
+                    (torch.exp(log_alpha_t - log_alpha_s)) * x
+                    - (sigma_t * phi_1) * model_s
+                    - (sigma_t * phi_2) * D1
+                    - (sigma_t * phi_3) * D2
+                )
+
+        if return_intermediate:
+            return x_t, {"model_s": model_s, "model_s1": model_s1, "model_s2": model_s2}
+        else:
+            return x_t
+
+    def multistep_dpm_solver_second_update(self, x, model_prev_list, t_prev_list, t, solver_type="dpmsolver"):
+        """
+        Multistep solver DPM-Solver-2 from time `t_prev_list[-1]` to time `t`.
+
+        Args:
+            x: A pytorch tensor. The initial value at time `s`.
+            model_prev_list: A list of pytorch tensor. The previous computed model values.
+            t_prev_list: A list of pytorch tensor. The previous times, each time has the shape (1,)
+            t: A pytorch tensor. The ending time, with the shape (1,).
+            solver_type: either 'dpmsolver' or 'taylor'. The type for the high-order solvers.
+                The type slightly impacts the performance. We recommend to use 'dpmsolver' type.
+        Returns:
+            x_t: A pytorch tensor. The approximated solution at time `t`.
+        """
+        if solver_type not in ["dpmsolver", "taylor"]:
+            raise ValueError(f"'solver_type' must be either 'dpmsolver' or 'taylor', got {solver_type}")
+        ns = self.noise_schedule
+        model_prev_1, model_prev_0 = model_prev_list[-2], model_prev_list[-1]
+        t_prev_1, t_prev_0 = t_prev_list[-2], t_prev_list[-1]
+        lambda_prev_1, lambda_prev_0, lambda_t = (
+            ns.marginal_lambda(t_prev_1),
+            ns.marginal_lambda(t_prev_0),
+            ns.marginal_lambda(t),
+        )
+        log_alpha_prev_0, log_alpha_t = ns.marginal_log_mean_coeff(t_prev_0), ns.marginal_log_mean_coeff(t)
+        sigma_prev_0, sigma_t = ns.marginal_std(t_prev_0), ns.marginal_std(t)
+        alpha_t = torch.exp(log_alpha_t)
+
+        h_0 = lambda_prev_0 - lambda_prev_1
+        h = lambda_t - lambda_prev_0
+        r0 = h_0 / h
+        D1_0 = (1.0 / r0) * (model_prev_0 - model_prev_1)
+        if self.algorithm_type == "dpmsolver++":
+            phi_1 = torch.expm1(-h)
+            if solver_type == "dpmsolver":
+                x_t = (sigma_t / sigma_prev_0) * x - (alpha_t * phi_1) * model_prev_0 - 0.5 * (alpha_t * phi_1) * D1_0
+            elif solver_type == "taylor":
+                x_t = (
+                    (sigma_t / sigma_prev_0) * x
+                    - (alpha_t * phi_1) * model_prev_0
+                    + (alpha_t * (phi_1 / h + 1.0)) * D1_0
+                )
+        else:
+            phi_1 = torch.expm1(h)
+            if solver_type == "dpmsolver":
+                x_t = (
+                    (torch.exp(log_alpha_t - log_alpha_prev_0)) * x
+                    - (sigma_t * phi_1) * model_prev_0
+                    - 0.5 * (sigma_t * phi_1) * D1_0
+                )
+            elif solver_type == "taylor":
+                x_t = (
+                    (torch.exp(log_alpha_t - log_alpha_prev_0)) * x
+                    - (sigma_t * phi_1) * model_prev_0
+                    - (sigma_t * (phi_1 / h - 1.0)) * D1_0
+                )
+        return x_t
+
+    def multistep_dpm_solver_third_update(self, x, model_prev_list, t_prev_list, t, solver_type="dpmsolver"):
+        """
+        Multistep solver DPM-Solver-3 from time `t_prev_list[-1]` to time `t`.
+
+        Args:
+            x: A pytorch tensor. The initial value at time `s`.
+            model_prev_list: A list of pytorch tensor. The previous computed model values.
+            t_prev_list: A list of pytorch tensor. The previous times, each time has the shape (1,)
+            t: A pytorch tensor. The ending time, with the shape (1,).
+            solver_type: either 'dpmsolver' or 'taylor'. The type for the high-order solvers.
+                The type slightly impacts the performance. We recommend to use 'dpmsolver' type.
+        Returns:
+            x_t: A pytorch tensor. The approximated solution at time `t`.
+        """
+        ns = self.noise_schedule
+        model_prev_2, model_prev_1, model_prev_0 = model_prev_list
+        t_prev_2, t_prev_1, t_prev_0 = t_prev_list
+        lambda_prev_2, lambda_prev_1, lambda_prev_0, lambda_t = (
+            ns.marginal_lambda(t_prev_2),
+            ns.marginal_lambda(t_prev_1),
+            ns.marginal_lambda(t_prev_0),
+            ns.marginal_lambda(t),
+        )
+        log_alpha_prev_0, log_alpha_t = ns.marginal_log_mean_coeff(t_prev_0), ns.marginal_log_mean_coeff(t)
+        sigma_prev_0, sigma_t = ns.marginal_std(t_prev_0), ns.marginal_std(t)
+        alpha_t = torch.exp(log_alpha_t)
+
+        h_1 = lambda_prev_1 - lambda_prev_2
+        h_0 = lambda_prev_0 - lambda_prev_1
+        h = lambda_t - lambda_prev_0
+        r0, r1 = h_0 / h, h_1 / h
+        D1_0 = (1.0 / r0) * (model_prev_0 - model_prev_1)
+        D1_1 = (1.0 / r1) * (model_prev_1 - model_prev_2)
+        D1 = D1_0 + (r0 / (r0 + r1)) * (D1_0 - D1_1)
+        D2 = (1.0 / (r0 + r1)) * (D1_0 - D1_1)
+        if self.algorithm_type == "dpmsolver++":
+            phi_1 = torch.expm1(-h)
+            phi_2 = phi_1 / h + 1.0
+            phi_3 = phi_2 / h - 0.5
+            x_t = (
+                (sigma_t / sigma_prev_0) * x
+                - (alpha_t * phi_1) * model_prev_0
+                + (alpha_t * phi_2) * D1
+                - (alpha_t * phi_3) * D2
+            )
+        else:
+            phi_1 = torch.expm1(h)
+            phi_2 = phi_1 / h - 1.0
+            phi_3 = phi_2 / h - 0.5
+            x_t = (
+                (torch.exp(log_alpha_t - log_alpha_prev_0)) * x
+                - (sigma_t * phi_1) * model_prev_0
+                - (sigma_t * phi_2) * D1
+                - (sigma_t * phi_3) * D2
+            )
+        return x_t
+
+    def singlestep_dpm_solver_update(
+        self, x, s, t, order, return_intermediate=False, solver_type="dpmsolver", r1=None, r2=None
+    ):
+        """
+        Singlestep DPM-Solver with the order `order` from time `s` to time `t`.
+
+        Args:
+            x: A pytorch tensor. The initial value at time `s`.
+            s: A pytorch tensor. The starting time, with the shape (1,).
+            t: A pytorch tensor. The ending time, with the shape (1,).
+            order: A `int`. The order of DPM-Solver. We only support order == 1 or 2 or 3.
+            return_intermediate: A `bool`. If true, also return the model value at time `s`, `s1` and `s2` (the intermediate times).
+            solver_type: either 'dpmsolver' or 'taylor'. The type for the high-order solvers.
+                The type slightly impacts the performance. We recommend to use 'dpmsolver' type.
+            r1: A `float`. The hyperparameter of the second-order or third-order solver.
+            r2: A `float`. The hyperparameter of the third-order solver.
+        Returns:
+            x_t: A pytorch tensor. The approximated solution at time `t`.
+        """
+        if order == 1:
+            return self.dpm_solver_first_update(x, s, t, return_intermediate=return_intermediate)
+        elif order == 2:
+            return self.singlestep_dpm_solver_second_update(
+                x, s, t, return_intermediate=return_intermediate, solver_type=solver_type, r1=r1
+            )
+        elif order == 3:
+            return self.singlestep_dpm_solver_third_update(
+                x, s, t, return_intermediate=return_intermediate, solver_type=solver_type, r1=r1, r2=r2
+            )
+        else:
+            raise ValueError(f"Solver order must be 1 or 2 or 3, got {order}")
+
+    def multistep_dpm_solver_update(self, x, model_prev_list, t_prev_list, t, order, solver_type="dpmsolver"):
+        """
+        Multistep DPM-Solver with the order `order` from time `t_prev_list[-1]` to time `t`.
+
+        Args:
+            x: A pytorch tensor. The initial value at time `s`.
+            model_prev_list: A list of pytorch tensor. The previous computed model values.
+            t_prev_list: A list of pytorch tensor. The previous times, each time has the shape (1,)
+            t: A pytorch tensor. The ending time, with the shape (1,).
+            order: A `int`. The order of DPM-Solver. We only support order == 1 or 2 or 3.
+            solver_type: either 'dpmsolver' or 'taylor'. The type for the high-order solvers.
+                The type slightly impacts the performance. We recommend to use 'dpmsolver' type.
+        Returns:
+            x_t: A pytorch tensor. The approximated solution at time `t`.
+        """
+        if order == 1:
+            return self.dpm_solver_first_update(x, t_prev_list[-1], t, model_s=model_prev_list[-1])
+        elif order == 2:
+            return self.multistep_dpm_solver_second_update(x, model_prev_list, t_prev_list, t, solver_type=solver_type)
+        elif order == 3:
+            return self.multistep_dpm_solver_third_update(x, model_prev_list, t_prev_list, t, solver_type=solver_type)
+        else:
+            raise ValueError(f"Solver order must be 1 or 2 or 3, got {order}")
+
+    def dpm_solver_adaptive(
+        self, x, order, t_T, t_0, h_init=0.05, atol=0.0078, rtol=0.05, theta=0.9, t_err=1e-5, solver_type="dpmsolver"
+    ):
+        """
+        The adaptive step size solver based on singlestep DPM-Solver.
+
+        Args:
+            x: A pytorch tensor. The initial value at time `t_T`.
+            order: A `int`. The (higher) order of the solver. We only support order == 2 or 3.
+            t_T: A `float`. The starting time of the sampling (default is T).
+            t_0: A `float`. The ending time of the sampling (default is epsilon).
+            h_init: A `float`. The initial step size (for logSNR).
+            atol: A `float`. The absolute tolerance of the solver. For image data, the default setting is 0.0078, followed [1].
+            rtol: A `float`. The relative tolerance of the solver. The default setting is 0.05.
+            theta: A `float`. The safety hyperparameter for adapting the step size. The default setting is 0.9, followed [1].
+            t_err: A `float`. The tolerance for the time. We solve the diffusion ODE until the absolute error between the
+                current time and `t_0` is less than `t_err`. The default setting is 1e-5.
+            solver_type: either 'dpmsolver' or 'taylor'. The type for the high-order solvers.
+                The type slightly impacts the performance. We recommend to use 'dpmsolver' type.
+        Returns:
+            x_0: A pytorch tensor. The approximated solution at time `t_0`.
+
+        [1] A. Jolicoeur-Martineau, K. Li, R. Piché-Taillefer, T. Kachman, and I. Mitliagkas, "Gotta go fast when generating data with score-based models," arXiv preprint arXiv:2105.14080, 2021.
+        """
+        ns = self.noise_schedule
+        s = t_T * torch.ones((1,)).to(x)
+        lambda_s = ns.marginal_lambda(s)
+        lambda_0 = ns.marginal_lambda(t_0 * torch.ones_like(s).to(x))
+        h = h_init * torch.ones_like(s).to(x)
+        x_prev = x
+        nfe = 0
+        if order == 2:
+            r1 = 0.5
+            lower_update = lambda x, s, t: self.dpm_solver_first_update(x, s, t, return_intermediate=True)
+            higher_update = lambda x, s, t, **kwargs: self.singlestep_dpm_solver_second_update(
+                x, s, t, r1=r1, solver_type=solver_type, **kwargs
+            )
+        elif order == 3:
+            r1, r2 = 1.0 / 3.0, 2.0 / 3.0
+            lower_update = lambda x, s, t: self.singlestep_dpm_solver_second_update(
+                x, s, t, r1=r1, return_intermediate=True, solver_type=solver_type
+            )
+            higher_update = lambda x, s, t, **kwargs: self.singlestep_dpm_solver_third_update(
+                x, s, t, r1=r1, r2=r2, solver_type=solver_type, **kwargs
+            )
+        else:
+            raise ValueError(f"For adaptive step size solver, order must be 2 or 3, got {order}")
+        while torch.abs(s - t_0).mean() > t_err:
+            t = ns.inverse_lambda(lambda_s + h)
+            x_lower, lower_noise_kwargs = lower_update(x, s, t)
+            x_higher = higher_update(x, s, t, **lower_noise_kwargs)
+            delta = torch.max(torch.ones_like(x).to(x) * atol, rtol * torch.max(torch.abs(x_lower), torch.abs(x_prev)))
+            norm_fn = lambda v: torch.sqrt(torch.square(v.reshape((v.shape[0], -1))).mean(dim=-1, keepdim=True))
+            E = norm_fn((x_higher - x_lower) / delta).max()
+            if torch.all(E <= 1.0):
+                x = x_higher
+                s = t
+                x_prev = x_lower
+                lambda_s = ns.marginal_lambda(s)
+            h = torch.min(theta * h * torch.float_power(E, -1.0 / order).float(), lambda_0 - lambda_s)
+            nfe += order
+        print("adaptive solver nfe", nfe)
+        return x
+
+    def add_noise(self, x, t, noise=None):
+        """
+        Compute the noised input xt = alpha_t * x + sigma_t * noise.
+
+        Args:
+            x: A `torch.Tensor` with shape `(batch_size, *shape)`.
+            t: A `torch.Tensor` with shape `(t_size,)`.
+        Returns:
+            xt with shape `(t_size, batch_size, *shape)`.
+        """
+        alpha_t, sigma_t = self.noise_schedule.marginal_alpha(t), self.noise_schedule.marginal_std(t)
+        if noise is None:
+            noise = torch.randn((t.shape[0], *x.shape), device=x.device)
+        x = x.reshape((-1, *x.shape))
+        xt = expand_dims(alpha_t, x.dim()) * x + expand_dims(sigma_t, x.dim()) * noise
+        if t.shape[0] == 1:
+            return xt.squeeze(0)
+        else:
+            return xt
+
+    def inverse(
+        self,
+        x,
+        steps=20,
+        t_start=None,
+        t_end=None,
+        order=2,
+        skip_type="time_uniform",
+        method="multistep",
+        lower_order_final=True,
+        denoise_to_zero=False,
+        solver_type="dpmsolver",
+        atol=0.0078,
+        rtol=0.05,
+        return_intermediate=False,
+    ):
+        """
+        Inverse the sample `x` from time `t_start` to `t_end` by DPM-Solver.
+        For discrete-time DPMs, we use `t_start=1/N`, where `N` is the total time steps during training.
+        """
+        t_0 = 1.0 / self.noise_schedule.total_N if t_start is None else t_start
+        t_T = self.noise_schedule.T if t_end is None else t_end
+        assert (
+            t_0 > 0 and t_T > 0
+        ), "Time range needs to be greater than 0. For discrete-time DPMs, it needs to be in [1 / N, 1], where N is the length of betas array"
+        return self.sample(
+            x,
+            steps=steps,
+            t_start=t_0,
+            t_end=t_T,
+            order=order,
+            skip_type=skip_type,
+            method=method,
+            lower_order_final=lower_order_final,
+            denoise_to_zero=denoise_to_zero,
+            solver_type=solver_type,
+            atol=atol,
+            rtol=rtol,
+            return_intermediate=return_intermediate,
+        )
+
+    def sample(
+        self,
+        x,
+        steps=20,
+        t_start=None,
+        t_end=None,
+        order=2,
+        skip_type="time_uniform",
+        method="multistep",
+        lower_order_final=True,
+        denoise_to_zero=False,
+        solver_type="dpmsolver",
+        atol=0.0078,
+        rtol=0.05,
+        return_intermediate=False,
+        flow_shift=1.0,
+    ):
+        """
+        Compute the sample at time `t_end` by DPM-Solver, given the initial `x` at time `t_start`.
+
+        =====================================================
+
+        We support the following algorithms for both noise prediction model and data prediction model:
+            - 'singlestep':
+                Singlestep DPM-Solver (i.e. "DPM-Solver-fast" in the paper), which combines different orders of singlestep DPM-Solver.
+                We combine all the singlestep solvers with order <= `order` to use up all the function evaluations (steps).
+                The total number of function evaluations (NFE) == `steps`.
+                Given a fixed NFE == `steps`, the sampling procedure is:
+                    - If `order` == 1:
+                        - Denote K = steps. We use K steps of DPM-Solver-1 (i.e. DDIM).
+                    - If `order` == 2:
+                        - Denote K = (steps // 2) + (steps % 2). We take K intermediate time steps for sampling.
+                        - If steps % 2 == 0, we use K steps of singlestep DPM-Solver-2.
+                        - If steps % 2 == 1, we use (K - 1) steps of singlestep DPM-Solver-2 and 1 step of DPM-Solver-1.
+                    - If `order` == 3:
+                        - Denote K = (steps // 3 + 1). We take K intermediate time steps for sampling.
+                        - If steps % 3 == 0, we use (K - 2) steps of singlestep DPM-Solver-3, and 1 step of singlestep DPM-Solver-2 and 1 step of DPM-Solver-1.
+                        - If steps % 3 == 1, we use (K - 1) steps of singlestep DPM-Solver-3 and 1 step of DPM-Solver-1.
+                        - If steps % 3 == 2, we use (K - 1) steps of singlestep DPM-Solver-3 and 1 step of singlestep DPM-Solver-2.
+            - 'multistep':
+                Multistep DPM-Solver with the order of `order`. The total number of function evaluations (NFE) == `steps`.
+                We initialize the first `order` values by lower order multistep solvers.
+                Given a fixed NFE == `steps`, the sampling procedure is:
+                    Denote K = steps.
+                    - If `order` == 1:
+                        - We use K steps of DPM-Solver-1 (i.e. DDIM).
+                    - If `order` == 2:
+                        - We firstly use 1 step of DPM-Solver-1, then use (K - 1) step of multistep DPM-Solver-2.
+                    - If `order` == 3:
+                        - We firstly use 1 step of DPM-Solver-1, then 1 step of multistep DPM-Solver-2, then (K - 2) step of multistep DPM-Solver-3.
+            - 'singlestep_fixed':
+                Fixed order singlestep DPM-Solver (i.e. DPM-Solver-1 or singlestep DPM-Solver-2 or singlestep DPM-Solver-3).
+                We use singlestep DPM-Solver-`order` for `order`=1 or 2 or 3, with total [`steps` // `order`] * `order` NFE.
+            - 'adaptive':
+                Adaptive step size DPM-Solver (i.e. "DPM-Solver-12" and "DPM-Solver-23" in the paper).
+                We ignore `steps` and use adaptive step size DPM-Solver with a higher order of `order`.
+                You can adjust the absolute tolerance `atol` and the relative tolerance `rtol` to balance the computatation costs
+                (NFE) and the sample quality.
+                    - If `order` == 2, we use DPM-Solver-12 which combines DPM-Solver-1 and singlestep DPM-Solver-2.
+                    - If `order` == 3, we use DPM-Solver-23 which combines singlestep DPM-Solver-2 and singlestep DPM-Solver-3.
+
+        =====================================================
+
+        Some advices for choosing the algorithm:
+            - For **unconditional sampling** or **guided sampling with small guidance scale** by DPMs:
+                Use singlestep DPM-Solver or DPM-Solver++ ("DPM-Solver-fast" in the paper) with `order = 3`.
+                e.g., DPM-Solver:
+                    >>> dpm_solver = DPM_Solver(model_fn, noise_schedule, algorithm_type="dpmsolver")
+                    >>> x_sample = dpm_solver.sample(x, steps=steps, t_start=t_start, t_end=t_end, order=3,
+                            skip_type='time_uniform', method='singlestep')
+                e.g., DPM-Solver++:
+                    >>> dpm_solver = DPM_Solver(model_fn, noise_schedule, algorithm_type="dpmsolver++")
+                    >>> x_sample = dpm_solver.sample(x, steps=steps, t_start=t_start, t_end=t_end, order=3,
+                            skip_type='time_uniform', method='singlestep')
+            - For **guided sampling with large guidance scale** by DPMs:
+                Use multistep DPM-Solver with `algorithm_type="dpmsolver++"` and `order = 2`.
+                e.g.
+                    >>> dpm_solver = DPM_Solver(model_fn, noise_schedule, algorithm_type="dpmsolver++")
+                    >>> x_sample = dpm_solver.sample(x, steps=steps, t_start=t_start, t_end=t_end, order=2,
+                            skip_type='time_uniform', method='multistep')
+
+        We support three types of `skip_type`:
+            - 'logSNR': uniform logSNR for the time steps. **Recommended for low-resolutional images**
+            - 'time_uniform': uniform time for the time steps. **Recommended for high-resolutional images**.
+            - 'time_quadratic': quadratic time for the time steps.
+
+        =====================================================
+        Args:
+            x: A pytorch tensor. The initial value at time `t_start`
+                e.g. if `t_start` == T, then `x` is a sample from the standard normal distribution.
+            steps: A `int`. The total number of function evaluations (NFE).
+            t_start: A `float`. The starting time of the sampling.
+                If `T` is None, we use self.noise_schedule.T (default is 1.0).
+            t_end: A `float`. The ending time of the sampling.
+                If `t_end` is None, we use 1. / self.noise_schedule.total_N.
+                e.g. if total_N == 1000, we have `t_end` == 1e-3.
+                For discrete-time DPMs:
+                    - We recommend `t_end` == 1. / self.noise_schedule.total_N.
+                For continuous-time DPMs:
+                    - We recommend `t_end` == 1e-3 when `steps` <= 15; and `t_end` == 1e-4 when `steps` > 15.
+            order: A `int`. The order of DPM-Solver.
+            skip_type: A `str`. The type for the spacing of the time steps. 'time_uniform' or 'logSNR' or 'time_quadratic'.
+            method: A `str`. The method for sampling. 'singlestep' or 'multistep' or 'singlestep_fixed' or 'adaptive'.
+            denoise_to_zero: A `bool`. Whether to denoise to time 0 at the final step.
+                Default is `False`. If `denoise_to_zero` is `True`, the total NFE is (`steps` + 1).
+
+                This trick is firstly proposed by DDPM (https://arxiv.org/abs/2006.11239) and
+                score_sde (https://arxiv.org/abs/2011.13456). Such trick can improve the FID
+                for diffusion models sampling by diffusion SDEs for low-resolutional images
+                (such as CIFAR-10). However, we observed that such trick does not matter for
+                high-resolutional images. As it needs an additional NFE, we do not recommend
+                it for high-resolutional images.
+            lower_order_final: A `bool`. Whether to use lower order solvers at the final steps.
+                Only valid for `method=multistep` and `steps < 15`. We empirically find that
+                this trick is a key to stabilizing the sampling by DPM-Solver with very few steps
+                (especially for steps <= 10). So we recommend to set it to be `True`.
+            solver_type: A `str`. The taylor expansion type for the solver. `dpmsolver` or `taylor`. We recommend `dpmsolver`.
+            atol: A `float`. The absolute tolerance of the adaptive step size solver. Valid when `method` == 'adaptive'.
+            rtol: A `float`. The relative tolerance of the adaptive step size solver. Valid when `method` == 'adaptive'.
+            return_intermediate: A `bool`. Whether to save the xt at each step.
+                When set to `True`, method returns a tuple (x0, intermediates); when set to False, method returns only x0.
+        Returns:
+            x_end: A pytorch tensor. The approximated solution at time `t_end`.
+
+        """
+        t_0 = 1.0 / self.noise_schedule.total_N if t_end is None else t_end
+        t_T = self.noise_schedule.T if t_start is None else t_start
+        assert (
+            t_0 > 0 and t_T > 0
+        ), "Time range needs to be greater than 0. For discrete-time DPMs, it needs to be in [1 / N, 1], where N is the length of betas array"
+        if return_intermediate:
+            assert method in [
+                "multistep",
+                "singlestep",
+                "singlestep_fixed",
+            ], "Cannot use adaptive solver when saving intermediate values"
+        if self.correcting_xt_fn is not None:
+            assert method in [
+                "multistep",
+                "singlestep",
+                "singlestep_fixed",
+            ], "Cannot use adaptive solver when correcting_xt_fn is not None"
+        device = x.device
+        intermediates = []
+        with torch.no_grad():
+            if method == "adaptive":
+                x = self.dpm_solver_adaptive(
+                    x, order=order, t_T=t_T, t_0=t_0, atol=atol, rtol=rtol, solver_type=solver_type
+                )
+            elif method == "multistep":
+                assert steps >= order
+                timesteps = self.get_time_steps(
+                    skip_type=skip_type, t_T=t_T, t_0=t_0, N=steps, device=device, shift=flow_shift
+                )
+                assert timesteps.shape[0] - 1 == steps
+                # Init the initial values.
+                step = 0
+                t = timesteps[step]
+                t_prev_list = [t]
+                model_prev_list = [self.model_fn(x, t)]
+                if self.correcting_xt_fn is not None:
+                    x = self.correcting_xt_fn(x, t, step)
+                if return_intermediate:
+                    intermediates.append(x)
+                self.update_progress(step + 1, len(timesteps))
+                # Init the first `order` values by lower order multistep DPM-Solver.
+                for step in range(1, order):
+                    t = timesteps[step]
+                    x = self.multistep_dpm_solver_update(
+                        x, model_prev_list, t_prev_list, t, step, solver_type=solver_type
+                    )
+                    if self.correcting_xt_fn is not None:
+                        x = self.correcting_xt_fn(x, t, step)
+                    if return_intermediate:
+                        intermediates.append(x)
+                    t_prev_list.append(t)
+                    model_prev_list.append(self.model_fn(x, t))
+                    # update progress bar
+                    self.update_progress(step + 1, len(timesteps))
+                # Compute the remaining values by `order`-th order multistep DPM-Solver.
+                for step in tqdm(range(order, steps + 1), disable=os.getenv("DPM_TQDM", "False") == "True"):
+                    t = timesteps[step]
+                    # We only use lower order for steps < 10
+                    # if lower_order_final and steps < 10:
+                    if lower_order_final:  # recommended by Shuchen Xue
+                        step_order = min(order, steps + 1 - step)
+                    else:
+                        step_order = order
+                    x = self.multistep_dpm_solver_update(
+                        x, model_prev_list, t_prev_list, t, step_order, solver_type=solver_type
+                    )
+                    if self.correcting_xt_fn is not None:
+                        x = self.correcting_xt_fn(x, t, step)
+                    if return_intermediate:
+                        intermediates.append(x)
+                    for i in range(order - 1):
+                        t_prev_list[i] = t_prev_list[i + 1]
+                        model_prev_list[i] = model_prev_list[i + 1]
+                    t_prev_list[-1] = t
+                    # We do not need to evaluate the final model value.
+                    if step < steps:
+                        model_prev_list[-1] = self.model_fn(x, t)
+                    # update progress bar
+                    self.update_progress(step + 1, len(timesteps))
+            elif method in ["singlestep", "singlestep_fixed"]:
+                if method == "singlestep":
+                    timesteps_outer, orders = self.get_orders_and_timesteps_for_singlestep_solver(
+                        steps=steps, order=order, skip_type=skip_type, t_T=t_T, t_0=t_0, device=device
+                    )
+                elif method == "singlestep_fixed":
+                    K = steps // order
+                    orders = [
+                        order,
+                    ] * K
+                    timesteps_outer = self.get_time_steps(skip_type=skip_type, t_T=t_T, t_0=t_0, N=K, device=device)
+                for step, order in enumerate(orders):
+                    s, t = timesteps_outer[step], timesteps_outer[step + 1]
+                    timesteps_inner = self.get_time_steps(
+                        skip_type=skip_type, t_T=s.item(), t_0=t.item(), N=order, device=device
+                    )
+                    lambda_inner = self.noise_schedule.marginal_lambda(timesteps_inner)
+                    h = lambda_inner[-1] - lambda_inner[0]
+                    r1 = None if order <= 1 else (lambda_inner[1] - lambda_inner[0]) / h
+                    r2 = None if order <= 2 else (lambda_inner[2] - lambda_inner[0]) / h
+                    x = self.singlestep_dpm_solver_update(x, s, t, order, solver_type=solver_type, r1=r1, r2=r2)
+                    if self.correcting_xt_fn is not None:
+                        x = self.correcting_xt_fn(x, t, step)
+                    if return_intermediate:
+                        intermediates.append(x)
+                    self.update_progress(step + 1, len(timesteps_outer))
+            else:
+                raise ValueError(f"Got wrong method {method}")
+            if denoise_to_zero:
+                t = torch.ones((1,)).to(device) * t_0
+                x = self.denoise_to_zero_fn(x, t)
+                if self.correcting_xt_fn is not None:
+                    x = self.correcting_xt_fn(x, t, step + 1)
+                if return_intermediate:
+                    intermediates.append(x)
+        if return_intermediate:
+            return x, intermediates
+        else:
+            return x
+
+    
+#############################################################
+# other utility functions
+#############################################################
+
+
+def interpolate_fn(x, xp, yp):
+    """
+    A piecewise linear function y = f(x), using xp and yp as keypoints.
+    We implement f(x) in a differentiable way (i.e. applicable for autograd).
+    The function f(x) is well-defined for all x-axis. (For x beyond the bounds of xp, we use the outmost points of xp to define the linear function.)
+
+    Args:
+        x: PyTorch tensor with shape [N, C], where N is the batch size, C is the number of channels (we use C = 1 for DPM-Solver).
+        xp: PyTorch tensor with shape [C, K], where K is the number of keypoints.
+        yp: PyTorch tensor with shape [C, K].
+    Returns:
+        The function values f(x), with shape [N, C].
+    """
+    N, K = x.shape[0], xp.shape[1]
+    all_x = torch.cat([x.unsqueeze(2), xp.unsqueeze(0).repeat((N, 1, 1))], dim=2)
+    sorted_all_x, x_indices = torch.sort(all_x, dim=2)
+    x_idx = torch.argmin(x_indices, dim=2)
+    cand_start_idx = x_idx - 1
+    start_idx = torch.where(
+        torch.eq(x_idx, 0),
+        torch.tensor(1, device=x.device),
+        torch.where(
+            torch.eq(x_idx, K),
+            torch.tensor(K - 2, device=x.device),
+            cand_start_idx,
+        ),
+    )
+    end_idx = torch.where(torch.eq(start_idx, cand_start_idx), start_idx + 2, start_idx + 1)
+    start_x = torch.gather(sorted_all_x, dim=2, index=start_idx.unsqueeze(2)).squeeze(2)
+    end_x = torch.gather(sorted_all_x, dim=2, index=end_idx.unsqueeze(2)).squeeze(2)
+    start_idx2 = torch.where(
+        torch.eq(x_idx, 0),
+        torch.tensor(0, device=x.device),
+        torch.where(
+            torch.eq(x_idx, K),
+            torch.tensor(K - 2, device=x.device),
+            cand_start_idx,
+        ),
+    )
+    y_positions_expanded = yp.unsqueeze(0).expand(N, -1, -1)
+    start_y = torch.gather(y_positions_expanded, dim=2, index=start_idx2.unsqueeze(2)).squeeze(2)
+    end_y = torch.gather(y_positions_expanded, dim=2, index=(start_idx2 + 1).unsqueeze(2)).squeeze(2)
+    cand = start_y + (x - start_x) * (end_y - start_y) / (end_x - start_x)
+    return cand
+
+
+def expand_dims(v, dims):
+    """
+    Expand the tensor `v` to the dim `dims`.
+
+    Args:
+        `v`: a PyTorch tensor with shape [N].
+        `dim`: a `int`.
+    Returns:
+        a PyTorch tensor with shape [N, 1, 1, ..., 1] and the total dimension is `dims`.
+    """
+    return v[(...,) + (None,) * (dims - 1)]

transport/integrators.py

@@ -0,0 +1,122 @@
+import torch as th
+from torchdiffeq import odeint
+from .utils import time_shift, get_lin_function
+
+class sde:
+    """SDE solver class"""
+
+    def __init__(
+        self,
+        drift,
+        diffusion,
+        *,
+        t0,
+        t1,
+        num_steps,
+        sampler_type,
+    ):
+        assert t0 < t1, "SDE sampler has to be in forward time"
+
+        self.num_timesteps = num_steps
+        self.t = th.linspace(t0, t1, num_steps)
+        self.dt = self.t[1] - self.t[0]
+        self.drift = drift
+        self.diffusion = diffusion
+        self.sampler_type = sampler_type
+
+    def __Euler_Maruyama_step(self, x, mean_x, t, model, **model_kwargs):
+        w_cur = th.randn(x.size()).to(x)
+        t = th.ones(x.size(0)).to(x) * t
+        dw = w_cur * th.sqrt(self.dt)
+        drift = self.drift(x, t, model, **model_kwargs)
+        diffusion = self.diffusion(x, t)
+        mean_x = x + drift * self.dt
+        x = mean_x + th.sqrt(2 * diffusion) * dw
+        return x, mean_x
+
+    def __Heun_step(self, x, _, t, model, **model_kwargs):
+        w_cur = th.randn(x.size()).to(x)
+        dw = w_cur * th.sqrt(self.dt)
+        t_cur = th.ones(x.size(0)).to(x) * t
+        diffusion = self.diffusion(x, t_cur)
+        xhat = x + th.sqrt(2 * diffusion) * dw
+        K1 = self.drift(xhat, t_cur, model, **model_kwargs)
+        xp = xhat + self.dt * K1
+        K2 = self.drift(xp, t_cur + self.dt, model, **model_kwargs)
+        return (
+            xhat + 0.5 * self.dt * (K1 + K2),
+            xhat,
+        )  # at last time point we do not perform the heun step
+
+    def __forward_fn(self):
+        """TODO: generalize here by adding all private functions ending with steps to it"""
+        sampler_dict = {
+            "Euler": self.__Euler_Maruyama_step,
+            "Heun": self.__Heun_step,
+        }
+
+        try:
+            sampler = sampler_dict[self.sampler_type]
+        except:
+            raise NotImplementedError("Smapler type not implemented.")
+
+        return sampler
+
+    def sample(self, init, model, **model_kwargs):
+        """forward loop of sde"""
+        x = init
+        mean_x = init
+        samples = []
+        sampler = self.__forward_fn()
+        for ti in self.t[:-1]:
+            with th.no_grad():
+                x, mean_x = sampler(x, mean_x, ti, model, **model_kwargs)
+                samples.append(x)
+
+        return samples
+
+
+class ode:
+    """ODE solver class"""
+
+    def __init__(
+        self,
+        drift,
+        *,
+        t0,
+        t1,
+        sampler_type,
+        num_steps,
+        atol,
+        rtol,
+        do_shift=False,
+        time_shifting_factor=None,
+    ):
+        assert t0 < t1, "ODE sampler has to be in forward time"
+
+        self.drift = drift
+        self.do_shift = do_shift
+        self.t = th.linspace(t0, t1, num_steps)
+        if time_shifting_factor:
+            self.t = self.t / (self.t + time_shifting_factor - time_shifting_factor * self.t)
+        self.atol = atol
+        self.rtol = rtol
+        self.sampler_type = sampler_type
+
+    def sample(self, x, model, **model_kwargs):
+        x = x.float()
+        device = x[0].device if isinstance(x, tuple) else x.device
+
+        def _fn(t, x):
+            t = th.ones(x[0].size(0)).to(device) * t if isinstance(x, tuple) else th.ones(x.size(0)).to(device) * t
+            model_output = self.drift(x, t, model, **model_kwargs).float()
+            return model_output
+        
+        t = self.t.to(device)
+        if self.do_shift:
+            mu = get_lin_function(y1=0.5, y2=1.15)(x.shape[1])
+            t = time_shift(mu, 1.0, t)
+        atol = [self.atol] * len(x) if isinstance(x, tuple) else [self.atol]
+        rtol = [self.rtol] * len(x) if isinstance(x, tuple) else [self.rtol]
+        samples = odeint(_fn, x, t, method=self.sampler_type, atol=atol, rtol=rtol)
+        return samples

transport/path.py

@@ -0,0 +1,201 @@
+import numpy as np
+import torch as th
+
+
+def expand_t_like_x(t, x):
+    """Function to reshape time t to broadcastable dimension of x
+    Args:
+      t: [batch_dim,], time vector
+      x: [batch_dim,...], data point
+    """
+    dims = [1] * len(x[0].size())
+    t = t.view(t.size(0), *dims)
+    return t
+
+
+#################### Coupling Plans ####################
+
+
+class ICPlan:
+    """Linear Coupling Plan"""
+
+    def __init__(self, sigma=0.0):
+        self.sigma = sigma
+
+    def compute_alpha_t(self, t):
+        """Compute the data coefficient along the path"""
+        return t, 1
+
+    def compute_sigma_t(self, t):
+        """Compute the noise coefficient along the path"""
+        return 1 - t, -1
+
+    def compute_d_alpha_alpha_ratio_t(self, t):
+        """Compute the ratio between d_alpha and alpha"""
+        return 1 / t
+
+    def compute_drift(self, x, t):
+        """We always output sde according to score parametrization;"""
+        t = expand_t_like_x(t, x)
+        alpha_ratio = self.compute_d_alpha_alpha_ratio_t(t)
+        sigma_t, d_sigma_t = self.compute_sigma_t(t)
+        drift = alpha_ratio * x
+        diffusion = alpha_ratio * (sigma_t**2) - sigma_t * d_sigma_t
+
+        return -drift, diffusion
+
+    def compute_diffusion(self, x, t, form="constant", norm=1.0):
+        """Compute the diffusion term of the SDE
+        Args:
+          x: [batch_dim, ...], data point
+          t: [batch_dim,], time vector
+          form: str, form of the diffusion term
+          norm: float, norm of the diffusion term
+        """
+        t = expand_t_like_x(t, x)
+        choices = {
+            "constant": norm,
+            "SBDM": norm * self.compute_drift(x, t)[1],
+            "sigma": norm * self.compute_sigma_t(t)[0],
+            "linear": norm * (1 - t),
+            "decreasing": 0.25 * (norm * th.cos(np.pi * t) + 1) ** 2,
+            "inccreasing-decreasing": norm * th.sin(np.pi * t) ** 2,
+        }
+
+        try:
+            diffusion = choices[form]
+        except KeyError:
+            raise NotImplementedError(f"Diffusion form {form} not implemented")
+
+        return diffusion
+
+    def get_score_from_velocity(self, velocity, x, t):
+        """Wrapper function: transfrom velocity prediction model to score
+        Args:
+            velocity: [batch_dim, ...] shaped tensor; velocity model output
+            x: [batch_dim, ...] shaped tensor; x_t data point
+            t: [batch_dim,] time tensor
+        """
+        t = expand_t_like_x(t, x)
+        alpha_t, d_alpha_t = self.compute_alpha_t(t)
+        sigma_t, d_sigma_t = self.compute_sigma_t(t)
+        mean = x
+        reverse_alpha_ratio = alpha_t / d_alpha_t
+        var = sigma_t**2 - reverse_alpha_ratio * d_sigma_t * sigma_t
+        score = (reverse_alpha_ratio * velocity - mean) / var
+        return score
+
+    def get_noise_from_velocity(self, velocity, x, t):
+        """Wrapper function: transfrom velocity prediction model to denoiser
+        Args:
+            velocity: [batch_dim, ...] shaped tensor; velocity model output
+            x: [batch_dim, ...] shaped tensor; x_t data point
+            t: [batch_dim,] time tensor
+        """
+        t = expand_t_like_x(t, x)
+        alpha_t, d_alpha_t = self.compute_alpha_t(t)
+        sigma_t, d_sigma_t = self.compute_sigma_t(t)
+        mean = x
+        reverse_alpha_ratio = alpha_t / d_alpha_t
+        var = reverse_alpha_ratio * d_sigma_t - sigma_t
+        noise = (reverse_alpha_ratio * velocity - mean) / var
+        return noise
+
+    def get_velocity_from_score(self, score, x, t):
+        """Wrapper function: transfrom score prediction model to velocity
+        Args:
+            score: [batch_dim, ...] shaped tensor; score model output
+            x: [batch_dim, ...] shaped tensor; x_t data point
+            t: [batch_dim,] time tensor
+        """
+        t = expand_t_like_x(t, x)
+        drift, var = self.compute_drift(x, t)
+        velocity = var * score - drift
+        return velocity
+
+    def compute_mu_t(self, t, x0, x1):
+        """Compute the mean of time-dependent density p_t"""
+        t = expand_t_like_x(t, x1)
+        alpha_t, _ = self.compute_alpha_t(t)
+        sigma_t, _ = self.compute_sigma_t(t)
+        if isinstance(x1, (list, tuple)):
+            return [alpha_t[i] * x1[i] + sigma_t[i] * x0[i] for i in range(len(x1))]
+        else:
+            return alpha_t * x1 + sigma_t * x0
+
+    def compute_xt(self, t, x0, x1):
+        """Sample xt from time-dependent density p_t; rng is required"""
+        xt = self.compute_mu_t(t, x0, x1)
+        return xt
+
+    def compute_ut(self, t, x0, x1, xt):
+        """Compute the vector field corresponding to p_t"""
+        t = expand_t_like_x(t, x1)
+        _, d_alpha_t = self.compute_alpha_t(t)
+        _, d_sigma_t = self.compute_sigma_t(t)
+        if isinstance(x1, (list, tuple)):
+            return [d_alpha_t * x1[i] + d_sigma_t * x0[i] for i in range(len(x1))]
+        else:
+            return d_alpha_t * x1 + d_sigma_t * x0
+
+    def plan(self, t, x0, x1):
+        xt = self.compute_xt(t, x0, x1)
+        ut = self.compute_ut(t, x0, x1, xt)
+        return t, xt, ut
+
+
+class VPCPlan(ICPlan):
+    """class for VP path flow matching"""
+
+    def __init__(self, sigma_min=0.1, sigma_max=20.0):
+        self.sigma_min = sigma_min
+        self.sigma_max = sigma_max
+        self.log_mean_coeff = (
+            lambda t: -0.25 * ((1 - t) ** 2) * (self.sigma_max - self.sigma_min) - 0.5 * (1 - t) * self.sigma_min
+        )
+        self.d_log_mean_coeff = lambda t: 0.5 * (1 - t) * (self.sigma_max - self.sigma_min) + 0.5 * self.sigma_min
+
+    def compute_alpha_t(self, t):
+        """Compute coefficient of x1"""
+        alpha_t = self.log_mean_coeff(t)
+        alpha_t = th.exp(alpha_t)
+        d_alpha_t = alpha_t * self.d_log_mean_coeff(t)
+        return alpha_t, d_alpha_t
+
+    def compute_sigma_t(self, t):
+        """Compute coefficient of x0"""
+        p_sigma_t = 2 * self.log_mean_coeff(t)
+        sigma_t = th.sqrt(1 - th.exp(p_sigma_t))
+        d_sigma_t = th.exp(p_sigma_t) * (2 * self.d_log_mean_coeff(t)) / (-2 * sigma_t)
+        return sigma_t, d_sigma_t
+
+    def compute_d_alpha_alpha_ratio_t(self, t):
+        """Special purposed function for computing numerical stabled d_alpha_t / alpha_t"""
+        return self.d_log_mean_coeff(t)
+
+    def compute_drift(self, x, t):
+        """Compute the drift term of the SDE"""
+        t = expand_t_like_x(t, x)
+        beta_t = self.sigma_min + (1 - t) * (self.sigma_max - self.sigma_min)
+        return -0.5 * beta_t * x, beta_t / 2
+
+
+class GVPCPlan(ICPlan):
+    def __init__(self, sigma=0.0):
+        super().__init__(sigma)
+
+    def compute_alpha_t(self, t):
+        """Compute coefficient of x1"""
+        alpha_t = th.sin(t * np.pi / 2)
+        d_alpha_t = np.pi / 2 * th.cos(t * np.pi / 2)
+        return alpha_t, d_alpha_t
+
+    def compute_sigma_t(self, t):
+        """Compute coefficient of x0"""
+        sigma_t = th.cos(t * np.pi / 2)
+        d_sigma_t = -np.pi / 2 * th.sin(t * np.pi / 2)
+        return sigma_t, d_sigma_t
+
+    def compute_d_alpha_alpha_ratio_t(self, t):
+        """Special purposed function for computing numerical stabled d_alpha_t / alpha_t"""
+        return np.pi / (2 * th.tan(t * np.pi / 2))

transport/transport.py

@@ -0,0 +1,490 @@
+import enum
+import math
+from typing import Callable
+
+import numpy as np
+import torch as th
+
+from . import path
+from .integrators import ode, sde
+from .utils import mean_flat, expand_dims
+from .dpm_solver import NoiseScheduleFlow, model_wrapper, DPM_Solver
+
+
+class ModelType(enum.Enum):
+    """
+    Which type of output the model predicts.
+    """
+
+    NOISE = enum.auto()  # the model predicts epsilon
+    SCORE = enum.auto()  # the model predicts \nabla \log p(x)
+    VELOCITY = enum.auto()  # the model predicts v(x)
+
+
+class PathType(enum.Enum):
+    """
+    Which type of path to use.
+    """
+
+    LINEAR = enum.auto()
+    GVP = enum.auto()
+    VP = enum.auto()
+
+
+class WeightType(enum.Enum):
+    """
+    Which type of weighting to use.
+    """
+
+    NONE = enum.auto()
+    VELOCITY = enum.auto()
+    LIKELIHOOD = enum.auto()
+
+
+class Transport:
+    def __init__(self, *, model_type, path_type, loss_type, train_eps, sample_eps, snr_type, do_shift, seq_len):
+        path_options = {
+            PathType.LINEAR: path.ICPlan,
+            PathType.GVP: path.GVPCPlan,
+            PathType.VP: path.VPCPlan,
+        }
+
+        self.loss_type = loss_type
+        self.model_type = model_type
+        self.path_sampler = path_options[path_type]()
+        self.train_eps = train_eps
+        self.sample_eps = sample_eps
+
+        self.snr_type = snr_type
+        self.do_shift = do_shift
+        self.seq_len = seq_len
+
+    def prior_logp(self, z):
+        """
+        Standard multivariate normal prior
+        Assume z is batched
+        """
+        shape = th.tensor(z.size())
+        N = th.prod(shape[1:])
+        _fn = lambda x: -N / 2.0 * np.log(2 * np.pi) - th.sum(x**2) / 2.0
+        return th.vmap(_fn)(z)
+
+    def check_interval(
+        self,
+        train_eps,
+        sample_eps,
+        *,
+        diffusion_form="SBDM",
+        sde=False,
+        reverse=False,
+        eval=False,
+        last_step_size=0.0,
+    ):
+        t0 = 0
+        t1 = 1
+        eps = train_eps if not eval else sample_eps
+        if type(self.path_sampler) in [path.VPCPlan]:
+            t1 = 1 - eps if (not sde or last_step_size == 0) else 1 - last_step_size
+
+        elif (type(self.path_sampler) in [path.ICPlan, path.GVPCPlan]) and (
+            self.model_type != ModelType.VELOCITY or sde
+        ):  # avoid numerical issue by taking a first semi-implicit step
+            t0 = eps if (diffusion_form == "SBDM" and sde) or self.model_type != ModelType.VELOCITY else 0
+            t1 = 1 - eps if (not sde or last_step_size == 0) else 1 - last_step_size
+
+        if reverse:
+            t0, t1 = 1 - t0, 1 - t1
+
+        return t0, t1
+
+    def sample(self, x1):
+        """Sampling x0 & t based on shape of x1 (if needed)
+        Args:
+          x1 - data point; [batch, *dim]
+        """
+        if isinstance(x1, (list, tuple)):
+            x0 = [th.randn_like(img_start) for img_start in x1]
+        else:
+            x0 = th.randn_like(x1)
+        t0, t1 = self.check_interval(self.train_eps, self.sample_eps)
+
+        if self.snr_type.startswith("uniform"):
+            assert t0 == 0.0 and t1 == 1.0, "not implemented."
+            if "_" in self.snr_type:
+                _, t0, t1 = self.snr_type.split("_")
+                t0, t1 = float(t0), float(t1)
+            t = th.rand((len(x1),)) * (t1 - t0) + t0
+        elif self.snr_type == "lognorm":
+            u = th.normal(mean=0.0, std=1.0, size=(len(x1),))
+            t = 1 / (1 + th.exp(-u)) * (t1 - t0) + t0
+        else:
+            raise NotImplementedError("Not implemented snr_type %s" % self.snr_type)
+
+        if self.do_shift:
+            base_shift: float = 0.5
+            max_shift: float = 1.15
+            mu = self.get_lin_function(y1=base_shift, y2=max_shift)(self.seq_len)
+            t = self.time_shift(mu, 1.0, t)
+        t = t.to(x1[0])
+        return t, x0, x1
+
+    def time_shift(self, mu: float, sigma: float, t: th.Tensor):
+        # the following implementation was original for t=0: clean / t=1: noise
+        # Since we adopt the reverse, the 1-t operations are needed
+        t = 1 - t
+        t = math.exp(mu) / (math.exp(mu) + (1 / t - 1) ** sigma)
+        t = 1 - t
+        return t
+
+    def get_lin_function(
+        self, x1: float = 256, y1: float = 0.5, x2: float = 4096, y2: float = 1.15
+    ) -> Callable[[float], float]:
+        m = (y2 - y1) / (x2 - x1)
+        b = y1 - m * x1
+        return lambda x: m * x + b
+
+    def training_losses(self, model, x1, model_kwargs=None):
+        """Loss for training the score model
+        Args:
+        - model: backbone model; could be score, noise, or velocity
+        - x1: datapoint
+        - model_kwargs: additional arguments for the model
+        """
+        if model_kwargs == None:
+            model_kwargs = {}
+        t, x0, x1 = self.sample(x1)
+        t, xt, ut = self.path_sampler.plan(t, x0, x1)
+        if "cond" in model_kwargs:
+            conds = model_kwargs.pop("cond")
+            xt = [th.cat([x, cond], dim=0) if cond is not None else x for x, cond in zip(xt, conds)]
+        model_output = model(xt, t, **model_kwargs)
+        B = len(x0)
+
+        terms = {}
+        # terms['pred'] = model_output
+        if self.model_type == ModelType.VELOCITY:
+            if isinstance(x1, (list, tuple)):
+                assert len(model_output) == len(ut) == len(x1)
+                for i in range(B):
+                    assert (
+                        model_output[i].shape == ut[i].shape == x1[i].shape
+                    ), f"{model_output[i].shape} {ut[i].shape} {x1[i].shape}"
+                terms["task_loss"] = th.stack(
+                    [((ut[i] - model_output[i]) ** 2).mean() for i in range(B)],
+                    dim=0,
+                )
+            else:
+                terms["task_loss"] = mean_flat(((model_output - ut) ** 2))
+        else:
+            raise NotImplementedError
+
+        terms["loss"] = terms["task_loss"]
+        terms["task_loss"] = terms["task_loss"].clone().detach()
+        terms["t"] = t
+        return terms
+
+    def get_drift(self):
+        """member function for obtaining the drift of the probability flow ODE"""
+
+        def score_ode(x, t, model, **model_kwargs):
+            drift_mean, drift_var = self.path_sampler.compute_drift(x, t)
+            model_output = model(x, t, **model_kwargs)
+            return -drift_mean + drift_var * model_output  # by change of variable
+
+        def noise_ode(x, t, model, **model_kwargs):
+            drift_mean, drift_var = self.path_sampler.compute_drift(x, t)
+            sigma_t, _ = self.path_sampler.compute_sigma_t(path.expand_t_like_x(t, x))
+            model_output = model(x, t, **model_kwargs)
+            score = model_output / -sigma_t
+            return -drift_mean + drift_var * score
+
+        def velocity_ode(x, t, model, **model_kwargs):
+            model_output = model(x, t, **model_kwargs)
+            return model_output
+
+        if self.model_type == ModelType.NOISE:
+            drift_fn = noise_ode
+        elif self.model_type == ModelType.SCORE:
+            drift_fn = score_ode
+        else:
+            drift_fn = velocity_ode
+
+        def body_fn(x, t, model, **model_kwargs):
+            model_output = drift_fn(x, t, model, **model_kwargs)
+            assert model_output.shape == x.shape, "Output shape from ODE solver must match input shape"
+            return model_output
+
+        return body_fn
+
+    def get_score(
+        self,
+    ):
+        """member function for obtaining score of
+        x_t = alpha_t * x + sigma_t * eps"""
+        if self.model_type == ModelType.NOISE:
+            score_fn = (
+                lambda x, t, model, **kwargs: model(x, t, **kwargs)
+                / -self.path_sampler.compute_sigma_t(path.expand_t_like_x(t, x))[0]
+            )
+        elif self.model_type == ModelType.SCORE:
+            score_fn = lambda x, t, model, **kwagrs: model(x, t, **kwagrs)
+        elif self.model_type == ModelType.VELOCITY:
+            score_fn = lambda x, t, model, **kwargs: self.path_sampler.get_score_from_velocity(
+                model(x, t, **kwargs), x, t
+            )
+        else:
+            raise NotImplementedError()
+
+        return score_fn
+
+
+class Sampler:
+    """Sampler class for the transport model"""
+
+    def __init__(
+        self,
+        transport,
+    ):
+        """Constructor for a general sampler; supporting different sampling methods
+        Args:
+        - transport: an tranport object specify model prediction & interpolant type
+        """
+
+        self.transport = transport
+        self.drift = self.transport.get_drift()
+        self.score = self.transport.get_score()
+
+    def __get_sde_diffusion_and_drift(
+        self,
+        *,
+        diffusion_form="SBDM",
+        diffusion_norm=1.0,
+    ):
+        def diffusion_fn(x, t):
+            diffusion = self.transport.path_sampler.compute_diffusion(x, t, form=diffusion_form, norm=diffusion_norm)
+            return diffusion
+
+        sde_drift = lambda x, t, model, **kwargs: self.drift(x, t, model, **kwargs) + diffusion_fn(x, t) * self.score(
+            x, t, model, **kwargs
+        )
+
+        sde_diffusion = diffusion_fn
+
+        return sde_drift, sde_diffusion
+
+    def __get_last_step(
+        self,
+        sde_drift,
+        *,
+        last_step,
+        last_step_size,
+    ):
+        """Get the last step function of the SDE solver"""
+
+        if last_step is None:
+            last_step_fn = lambda x, t, model, **model_kwargs: x
+        elif last_step == "Mean":
+            last_step_fn = (
+                lambda x, t, model, **model_kwargs: x + sde_drift(x, t, model, **model_kwargs) * last_step_size
+            )
+        elif last_step == "Tweedie":
+            alpha = self.transport.path_sampler.compute_alpha_t  # simple aliasing; the original name was too long
+            sigma = self.transport.path_sampler.compute_sigma_t
+            last_step_fn = lambda x, t, model, **model_kwargs: x / alpha(t)[0][0] + (sigma(t)[0][0] ** 2) / alpha(t)[0][
+                0
+            ] * self.score(x, t, model, **model_kwargs)
+        elif last_step == "Euler":
+            last_step_fn = (
+                lambda x, t, model, **model_kwargs: x + self.drift(x, t, model, **model_kwargs) * last_step_size
+            )
+        else:
+            raise NotImplementedError()
+
+        return last_step_fn
+
+    def sample_sde(
+        self,
+        *,
+        sampling_method="Euler",
+        diffusion_form="SBDM",
+        diffusion_norm=1.0,
+        last_step="Mean",
+        last_step_size=0.04,
+        num_steps=250,
+    ):
+        """returns a sampling function with given SDE settings
+        Args:
+        - sampling_method: type of sampler used in solving the SDE; default to be Euler-Maruyama
+        - diffusion_form: function form of diffusion coefficient; default to be matching SBDM
+        - diffusion_norm: function magnitude of diffusion coefficient; default to 1
+        - last_step: type of the last step; default to identity
+        - last_step_size: size of the last step; default to match the stride of 250 steps over [0,1]
+        - num_steps: total integration step of SDE
+        """
+
+        if last_step is None:
+            last_step_size = 0.0
+
+        sde_drift, sde_diffusion = self.__get_sde_diffusion_and_drift(
+            diffusion_form=diffusion_form,
+            diffusion_norm=diffusion_norm,
+        )
+
+        t0, t1 = self.transport.check_interval(
+            self.transport.train_eps,
+            self.transport.sample_eps,
+            diffusion_form=diffusion_form,
+            sde=True,
+            eval=True,
+            reverse=False,
+            last_step_size=last_step_size,
+        )
+
+        _sde = sde(
+            sde_drift,
+            sde_diffusion,
+            t0=t0,
+            t1=t1,
+            num_steps=num_steps,
+            sampler_type=sampling_method,
+        )
+
+        last_step_fn = self.__get_last_step(sde_drift, last_step=last_step, last_step_size=last_step_size)
+
+        def _sample(init, model, **model_kwargs):
+            xs = _sde.sample(init, model, **model_kwargs)
+            ts = th.ones(init.size(0), device=init.device) * t1
+            x = last_step_fn(xs[-1], ts, model, **model_kwargs)
+            xs.append(x)
+
+            assert len(xs) == num_steps, "Samples does not match the number of steps"
+
+            return xs
+
+        return _sample
+    
+    def sample_dpm(
+        self,
+        model,
+        model_kwargs=None,
+    ):
+
+        noise_schedule = NoiseScheduleFlow(schedule="discrete_flow")
+
+        def noise_pred_fn(x, t_continuous):
+            output = model(x, 1 - t_continuous, **model_kwargs)
+            _, sigma_t = noise_schedule.marginal_alpha(t_continuous), noise_schedule.marginal_std(t_continuous)
+            try:
+                noise = x - (1 - expand_dims(sigma_t, x.dim()).to(x)) * output
+            except:
+                noise = x - (1 - expand_dims(sigma_t, x.dim()).to(x)) * output[0]
+            return noise
+
+        return DPM_Solver(noise_pred_fn, noise_schedule, algorithm_type="dpmsolver++").sample
+
+
+    def sample_ode(
+        self,
+        *,
+        sampling_method="dopri5",
+        num_steps=50,
+        atol=1e-6,
+        rtol=1e-3,
+        reverse=False,
+        do_shift=False,
+        time_shifting_factor=None, 
+    ):
+        """returns a sampling function with given ODE settings
+        Args:
+        - sampling_method: type of sampler used in solving the ODE; default to be Dopri5
+        - num_steps:
+            - fixed solver (Euler, Heun): the actual number of integration steps performed
+            - adaptive solver (Dopri5): the number of datapoints saved during integration; produced by interpolation
+        - atol: absolute error tolerance for the solver
+        - rtol: relative error tolerance for the solver
+        """
+
+        # for flux
+        drift = lambda x, t, model, **kwargs: self.drift(x, t, model, **kwargs)
+
+        t0, t1 = self.transport.check_interval(
+            self.transport.train_eps,
+            self.transport.sample_eps,
+            sde=False,
+            eval=True,
+            reverse=reverse,
+            last_step_size=0.0,
+        )
+
+        _ode = ode(
+            drift=drift,
+            t0=t0,
+            t1=t1,
+            sampler_type=sampling_method,
+            num_steps=num_steps,
+            atol=atol,
+            rtol=rtol,
+            do_shift=do_shift,
+            time_shifting_factor=time_shifting_factor,
+        )
+
+        return _ode.sample
+
+    def sample_ode_likelihood(
+        self,
+        *,
+        sampling_method="dopri5",
+        num_steps=50,
+        atol=1e-6,
+        rtol=1e-3,
+    ):
+        """returns a sampling function for calculating likelihood with given ODE settings
+        Args:
+        - sampling_method: type of sampler used in solving the ODE; default to be Dopri5
+        - num_steps:
+            - fixed solver (Euler, Heun): the actual number of integration steps performed
+            - adaptive solver (Dopri5): the number of datapoints saved during integration; produced by interpolation
+        - atol: absolute error tolerance for the solver
+        - rtol: relative error tolerance for the solver
+        """
+
+        def _likelihood_drift(x, t, model, **model_kwargs):
+            x, _ = x
+            eps = th.randint(2, x.size(), dtype=th.float, device=x.device) * 2 - 1
+            t = th.ones_like(t) * (1 - t)
+            with th.enable_grad():
+                x.requires_grad = True
+                grad = th.autograd.grad(th.sum(self.drift(x, t, model, **model_kwargs) * eps), x)[0]
+                logp_grad = th.sum(grad * eps, dim=tuple(range(1, len(x.size()))))
+                drift = self.drift(x, t, model, **model_kwargs)
+            return (-drift, logp_grad)
+
+        t0, t1 = self.transport.check_interval(
+            self.transport.train_eps,
+            self.transport.sample_eps,
+            sde=False,
+            eval=True,
+            reverse=False,
+            last_step_size=0.0,
+        )
+
+        _ode = ode(
+            drift=_likelihood_drift,
+            t0=t0,
+            t1=t1,
+            sampler_type=sampling_method,
+            num_steps=num_steps,
+            atol=atol,
+            rtol=rtol,
+        )
+
+        def _sample_fn(x, model, **model_kwargs):
+            init_logp = th.zeros(x.size(0)).to(x)
+            input = (x, init_logp)
+            drift, delta_logp = _ode.sample(input, model, **model_kwargs)
+            drift, delta_logp = drift[-1], delta_logp[-1]
+            prior_logp = self.transport.prior_logp(drift)
+            logp = prior_logp - delta_logp
+            return logp, drift
+
+        return _sample_fn

transport/utils.py

@@ -0,0 +1,56 @@
+import torch as th
+import math
+
+class EasyDict:
+    def __init__(self, sub_dict):
+        for k, v in sub_dict.items():
+            setattr(self, k, v)
+
+    def __getitem__(self, key):
+        return getattr(self, key)
+
+
+def mean_flat(x):
+    """
+    Take the mean over all non-batch dimensions.
+    """
+    return th.mean(x, dim=list(range(1, len(x.size()))))
+
+
+def log_state(state):
+    result = []
+
+    sorted_state = dict(sorted(state.items()))
+    for key, value in sorted_state.items():
+        # Check if the value is an instance of a class
+        if "<object" in str(value) or "object at" in str(value):
+            result.append(f"{key}: [{value.__class__.__name__}]")
+        else:
+            result.append(f"{key}: {value}")
+
+    return "\n".join(result)
+
+def time_shift(mu: float, sigma: float, t: th.Tensor):
+    # the following implementation was original for t=0: clean / t=1: noise
+    # Since we adopt the reverse, the 1-t operations are needed
+    t = 1 - t
+    t = math.exp(mu) / (math.exp(mu) + (1 / t - 1) ** sigma)
+    t = 1 - t
+    return t
+
+def get_lin_function(x1: float = 256, y1: float = 0.5, x2: float = 4096, y2: float = 1.15):
+    m = (y2 - y1) / (x2 - x1)
+    b = y1 - m * x1
+    return lambda x: m * x + b
+
+def expand_dims(v, dims):
+    """
+    Expand the tensor `v` to the dim `dims`.
+
+    Args:
+        `v`: a PyTorch tensor with shape [N].
+        `dim`: a `int`.
+    Returns:
+        a PyTorch tensor with shape [N, 1, 1, ..., 1] and the total dimension is `dims`.
+    """
+    return v[(...,) + (None,) * (dims - 1)]