scheduling_dpmsolver_singlestep.py

# Copyright 2023 TSAIL Team and The HuggingFace Team. All rights reserved.
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
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#     http://www.apache.org/licenses/LICENSE-2.0
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# Unless required by applicable law or agreed to in writing, software
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# DISCLAIMER: This file is strongly influenced by https://github.com/LuChengTHU/dpm-solver

import math
from typing import List, Optional, Tuple, Union

import numpy as np
import torch

from ..configuration_utils import ConfigMixin, register_to_config
from .scheduling_utils import KarrasDiffusionSchedulers, SchedulerMixin, SchedulerOutput


# Copied from diffusers.schedulers.scheduling_ddpm.betas_for_alpha_bar
def betas_for_alpha_bar(num_diffusion_timesteps, max_beta=0.999):
    """
    Create a beta schedule that discretizes the given alpha_t_bar function, which defines the cumulative product of
    (1-beta) over time from t = [0,1].

    Contains a function alpha_bar that takes an argument t and transforms it to the cumulative product of (1-beta) up
    to that part of the diffusion process.


    Args:
        num_diffusion_timesteps (`int`): the number of betas to produce.
        max_beta (`float`): the maximum beta to use; use values lower than 1 to
                     prevent singularities.

    Returns:
        betas (`np.ndarray`): the betas used by the scheduler to step the model outputs
    """

    def alpha_bar(time_step):
        return math.cos((time_step + 0.008) / 1.008 * math.pi / 2) ** 2

    betas = []
    for i in range(num_diffusion_timesteps):
        t1 = i / num_diffusion_timesteps
        t2 = (i + 1) / num_diffusion_timesteps
        betas.append(min(1 - alpha_bar(t2) / alpha_bar(t1), max_beta))
    return torch.tensor(betas, dtype=torch.float32)


class DPMSolverSinglestepScheduler(SchedulerMixin, ConfigMixin):
    """
    DPM-Solver (and the improved version DPM-Solver++) is a fast dedicated high-order solver for diffusion ODEs with
    the convergence order guarantee. Empirically, sampling by DPM-Solver with only 20 steps can generate high-quality
    samples, and it can generate quite good samples even in only 10 steps.

    For more details, see the original paper: https://arxiv.org/abs/2206.00927 and https://arxiv.org/abs/2211.01095

    Currently, we support the singlestep DPM-Solver for both noise prediction models and data prediction models. We
    recommend to use `solver_order=2` for guided sampling, and `solver_order=3` for unconditional sampling.

    We also support the "dynamic thresholding" method in Imagen (https://arxiv.org/abs/2205.11487). For pixel-space
    diffusion models, you can set both `algorithm_type="dpmsolver++"` and `thresholding=True` to use the dynamic
    thresholding. Note that the thresholding method is unsuitable for latent-space diffusion models (such as
    stable-diffusion).

    [`~ConfigMixin`] takes care of storing all config attributes that are passed in the scheduler's `__init__`
    function, such as `num_train_timesteps`. They can be accessed via `scheduler.config.num_train_timesteps`.
    [`SchedulerMixin`] provides general loading and saving functionality via the [`SchedulerMixin.save_pretrained`] and
    [`~SchedulerMixin.from_pretrained`] functions.

    Args:
        num_train_timesteps (`int`): number of diffusion steps used to train the model.
        beta_start (`float`): the starting `beta` value of inference.
        beta_end (`float`): the final `beta` value.
        beta_schedule (`str`):
            the beta schedule, a mapping from a beta range to a sequence of betas for stepping the model. Choose from
            `linear`, `scaled_linear`, or `squaredcos_cap_v2`.
        trained_betas (`np.ndarray`, optional):
            option to pass an array of betas directly to the constructor to bypass `beta_start`, `beta_end` etc.
        solver_order (`int`, default `2`):
            the order of DPM-Solver; can be `1` or `2` or `3`. We recommend to use `solver_order=2` for guided
            sampling, and `solver_order=3` for unconditional sampling.
        prediction_type (`str`, default `epsilon`):
            indicates whether the model predicts the noise (epsilon), or the data / `x0`. One of `epsilon`, `sample`,
            or `v-prediction`.
        thresholding (`bool`, default `False`):
            whether to use the "dynamic thresholding" method (introduced by Imagen, https://arxiv.org/abs/2205.11487).
            For pixel-space diffusion models, you can set both `algorithm_type=dpmsolver++` and `thresholding=True` to
            use the dynamic thresholding. Note that the thresholding method is unsuitable for latent-space diffusion
            models (such as stable-diffusion).
        dynamic_thresholding_ratio (`float`, default `0.995`):
            the ratio for the dynamic thresholding method. Default is `0.995`, the same as Imagen
            (https://arxiv.org/abs/2205.11487).
        sample_max_value (`float`, default `1.0`):
            the threshold value for dynamic thresholding. Valid only when `thresholding=True` and
            `algorithm_type="dpmsolver++`.
        algorithm_type (`str`, default `dpmsolver++`):
            the algorithm type for the solver. Either `dpmsolver` or `dpmsolver++`. The `dpmsolver` type implements the
            algorithms in https://arxiv.org/abs/2206.00927, and the `dpmsolver++` type implements the algorithms in
            https://arxiv.org/abs/2211.01095. We recommend to use `dpmsolver++` with `solver_order=2` for guided
            sampling (e.g. stable-diffusion).
        solver_type (`str`, default `midpoint`):
            the solver type for the second-order solver. Either `midpoint` or `heun`. The solver type slightly affects
            the sample quality, especially for small number of steps. We empirically find that `midpoint` solvers are
            slightly better, so we recommend to use the `midpoint` type.
        lower_order_final (`bool`, default `True`):
            whether to use lower-order solvers in the final steps. For singlestep schedulers, we recommend to enable
            this to use up all the function evaluations.
        lambda_min_clipped (`float`, default `-inf`):
            the clipping threshold for the minimum value of lambda(t) for numerical stability. This is critical for
            cosine (squaredcos_cap_v2) noise schedule.
        variance_type (`str`, *optional*):
            Set to "learned" or "learned_range" for diffusion models that predict variance. For example, OpenAI's
            guided-diffusion (https://github.com/openai/guided-diffusion) predicts both mean and variance of the
            Gaussian distribution in the model's output. DPM-Solver only needs the "mean" output because it is based on
            diffusion ODEs. whether the model's output contains the predicted Gaussian variance. For example, OpenAI's
            guided-diffusion (https://github.com/openai/guided-diffusion) predicts both mean and variance of the
            Gaussian distribution in the model's output. DPM-Solver only needs the "mean" output because it is based on
            diffusion ODEs.

    """

    _compatibles = [e.name for e in KarrasDiffusionSchedulers]
    order = 1

    @register_to_config
    def __init__(
        self,
        num_train_timesteps: int = 1000,
        beta_start: float = 0.0001,
        beta_end: float = 0.02,
        beta_schedule: str = "linear",
        trained_betas: Optional[np.ndarray] = None,
        solver_order: int = 2,
        prediction_type: str = "epsilon",
        thresholding: bool = False,
        dynamic_thresholding_ratio: float = 0.995,
        sample_max_value: float = 1.0,
        algorithm_type: str = "dpmsolver++",
        solver_type: str = "midpoint",
        lower_order_final: bool = True,
        lambda_min_clipped: float = -float("inf"),
        variance_type: Optional[str] = None,
    ):
        if trained_betas is not None:
            self.betas = torch.tensor(trained_betas, dtype=torch.float32)
        elif beta_schedule == "linear":
            self.betas = torch.linspace(beta_start, beta_end, num_train_timesteps, dtype=torch.float32)
        elif beta_schedule == "scaled_linear":
            # this schedule is very specific to the latent diffusion model.
            self.betas = (
                torch.linspace(beta_start**0.5, beta_end**0.5, num_train_timesteps, dtype=torch.float32) ** 2
            )
        elif beta_schedule == "squaredcos_cap_v2":
            # Glide cosine schedule
            self.betas = betas_for_alpha_bar(num_train_timesteps)
        else:
            raise NotImplementedError(f"{beta_schedule} does is not implemented for {self.__class__}")

        self.alphas = 1.0 - self.betas
        self.alphas_cumprod = torch.cumprod(self.alphas, dim=0)
        # Currently we only support VP-type noise schedule
        self.alpha_t = torch.sqrt(self.alphas_cumprod)
        self.sigma_t = torch.sqrt(1 - self.alphas_cumprod)
        self.lambda_t = torch.log(self.alpha_t) - torch.log(self.sigma_t)

        # standard deviation of the initial noise distribution
        self.init_noise_sigma = 1.0

        # settings for DPM-Solver
        if algorithm_type not in ["dpmsolver", "dpmsolver++"]:
            if algorithm_type == "deis":
                self.register_to_config(algorithm_type="dpmsolver++")
            else:
                raise NotImplementedError(f"{algorithm_type} does is not implemented for {self.__class__}")
        if solver_type not in ["midpoint", "heun"]:
            if solver_type in ["logrho", "bh1", "bh2"]:
                self.register_to_config(solver_type="midpoint")
            else:
                raise NotImplementedError(f"{solver_type} does is not implemented for {self.__class__}")

        # setable values
        self.num_inference_steps = None
        timesteps = np.linspace(0, num_train_timesteps - 1, num_train_timesteps, dtype=np.float32)[::-1].copy()
        self.timesteps = torch.from_numpy(timesteps)
        self.model_outputs = [None] * solver_order
        self.sample = None
        self.order_list = self.get_order_list(num_train_timesteps)

    def get_order_list(self, num_inference_steps: int) -> List[int]:
        """
        Computes the solver order at each time step.

        Args:
            num_inference_steps (`int`):
                the number of diffusion steps used when generating samples with a pre-trained model.
        """
        steps = num_inference_steps
        order = self.config.solver_order
        if self.config.lower_order_final:
            if order == 3:
                if steps % 3 == 0:
                    orders = [1, 2, 3] * (steps // 3 - 1) + [1, 2] + [1]
                elif steps % 3 == 1:
                    orders = [1, 2, 3] * (steps // 3) + [1]
                else:
                    orders = [1, 2, 3] * (steps // 3) + [1, 2]
            elif order == 2:
                if steps % 2 == 0:
                    orders = [1, 2] * (steps // 2)
                else:
                    orders = [1, 2] * (steps // 2) + [1]
            elif order == 1:
                orders = [1] * steps
        else:
            if order == 3:
                orders = [1, 2, 3] * (steps // 3)
            elif order == 2:
                orders = [1, 2] * (steps // 2)
            elif order == 1:
                orders = [1] * steps
        return orders

    def set_timesteps(self, num_inference_steps: int, device: Union[str, torch.device] = None):
        """
        Sets the timesteps used for the diffusion chain. Supporting function to be run before inference.

        Args:
            num_inference_steps (`int`):
                the number of diffusion steps used when generating samples with a pre-trained model.
            device (`str` or `torch.device`, optional):
                the device to which the timesteps should be moved to. If `None`, the timesteps are not moved.
        """
        self.num_inference_steps = num_inference_steps
        # Clipping the minimum of all lambda(t) for numerical stability.
        # This is critical for cosine (squaredcos_cap_v2) noise schedule.
        clipped_idx = torch.searchsorted(torch.flip(self.lambda_t, [0]), self.config.lambda_min_clipped)
        timesteps = (
            np.linspace(0, self.config.num_train_timesteps - 1 - clipped_idx, num_inference_steps + 1)
            .round()[::-1][:-1]
            .copy()
            .astype(np.int64)
        )
        self.timesteps = torch.from_numpy(timesteps).to(device)
        self.model_outputs = [None] * self.config.solver_order
        self.sample = None
        self.orders = self.get_order_list(num_inference_steps)

    # Copied from diffusers.schedulers.scheduling_ddpm.DDPMScheduler._threshold_sample
    def _threshold_sample(self, sample: torch.FloatTensor) -> torch.FloatTensor:
        """
        "Dynamic thresholding: At each sampling step we set s to a certain percentile absolute pixel value in xt0 (the
        prediction of x_0 at timestep t), and if s > 1, then we threshold xt0 to the range [-s, s] and then divide by
        s. Dynamic thresholding pushes saturated pixels (those near -1 and 1) inwards, thereby actively preventing
        pixels from saturation at each step. We find that dynamic thresholding results in significantly better
        photorealism as well as better image-text alignment, especially when using very large guidance weights."

        https://arxiv.org/abs/2205.11487
        """
        dtype = sample.dtype
        batch_size, channels, height, width = sample.shape

        if dtype not in (torch.float32, torch.float64):
            sample = sample.float()  # upcast for quantile calculation, and clamp not implemented for cpu half

        # Flatten sample for doing quantile calculation along each image
        sample = sample.reshape(batch_size, channels * height * width)

        abs_sample = sample.abs()  # "a certain percentile absolute pixel value"

        s = torch.quantile(abs_sample, self.config.dynamic_thresholding_ratio, dim=1)
        s = torch.clamp(
            s, min=1, max=self.config.sample_max_value
        )  # When clamped to min=1, equivalent to standard clipping to [-1, 1]

        s = s.unsqueeze(1)  # (batch_size, 1) because clamp will broadcast along dim=0
        sample = torch.clamp(sample, -s, s) / s  # "we threshold xt0 to the range [-s, s] and then divide by s"

        sample = sample.reshape(batch_size, channels, height, width)
        sample = sample.to(dtype)

        return sample

    def convert_model_output(
        self, model_output: torch.FloatTensor, timestep: int, sample: torch.FloatTensor
    ) -> torch.FloatTensor:
        """
        Convert the model output to the corresponding type that the algorithm (DPM-Solver / DPM-Solver++) needs.

        DPM-Solver is designed to discretize an integral of the noise prediction model, and DPM-Solver++ is designed to
        discretize an integral of the data prediction model. So we need to first convert the model output to the
        corresponding type to match the algorithm.

        Note that the algorithm type and the model type is decoupled. That is to say, we can use either DPM-Solver or
        DPM-Solver++ for both noise prediction model and data prediction model.

        Args:
            model_output (`torch.FloatTensor`): direct output from learned diffusion model.
            timestep (`int`): current discrete timestep in the diffusion chain.
            sample (`torch.FloatTensor`):
                current instance of sample being created by diffusion process.

        Returns:
            `torch.FloatTensor`: the converted model output.
        """
        # DPM-Solver++ needs to solve an integral of the data prediction model.
        if self.config.algorithm_type == "dpmsolver++":
            if self.config.prediction_type == "epsilon":
                # DPM-Solver and DPM-Solver++ only need the "mean" output.
                if self.config.variance_type in ["learned_range"]:
                    model_output = model_output[:, :3]
                alpha_t, sigma_t = self.alpha_t[timestep], self.sigma_t[timestep]
                x0_pred = (sample - sigma_t * model_output) / alpha_t
            elif self.config.prediction_type == "sample":
                x0_pred = model_output
            elif self.config.prediction_type == "v_prediction":
                alpha_t, sigma_t = self.alpha_t[timestep], self.sigma_t[timestep]
                x0_pred = alpha_t * sample - sigma_t * model_output
            else:
                raise ValueError(
                    f"prediction_type given as {self.config.prediction_type} must be one of `epsilon`, `sample`, or"
                    " `v_prediction` for the DPMSolverSinglestepScheduler."
                )

            if self.config.thresholding:
                x0_pred = self._threshold_sample(x0_pred)

            return x0_pred
        # DPM-Solver needs to solve an integral of the noise prediction model.
        elif self.config.algorithm_type == "dpmsolver":
            if self.config.prediction_type == "epsilon":
                # DPM-Solver and DPM-Solver++ only need the "mean" output.
                if self.config.variance_type in ["learned_range"]:
                    model_output = model_output[:, :3]
                return model_output
            elif self.config.prediction_type == "sample":
                alpha_t, sigma_t = self.alpha_t[timestep], self.sigma_t[timestep]
                epsilon = (sample - alpha_t * model_output) / sigma_t
                return epsilon
            elif self.config.prediction_type == "v_prediction":
                alpha_t, sigma_t = self.alpha_t[timestep], self.sigma_t[timestep]
                epsilon = alpha_t * model_output + sigma_t * sample
                return epsilon
            else:
                raise ValueError(
                    f"prediction_type given as {self.config.prediction_type} must be one of `epsilon`, `sample`, or"
                    " `v_prediction` for the DPMSolverSinglestepScheduler."
                )

    def dpm_solver_first_order_update(
        self,
        model_output: torch.FloatTensor,
        timestep: int,
        prev_timestep: int,
        sample: torch.FloatTensor,
    ) -> torch.FloatTensor:
        """
        One step for the first-order DPM-Solver (equivalent to DDIM).

        See https://arxiv.org/abs/2206.00927 for the detailed derivation.

        Args:
            model_output (`torch.FloatTensor`): direct output from learned diffusion model.
            timestep (`int`): current discrete timestep in the diffusion chain.
            prev_timestep (`int`): previous discrete timestep in the diffusion chain.
            sample (`torch.FloatTensor`):
                current instance of sample being created by diffusion process.

        Returns:
            `torch.FloatTensor`: the sample tensor at the previous timestep.
        """
        lambda_t, lambda_s = self.lambda_t[prev_timestep], self.lambda_t[timestep]
        alpha_t, alpha_s = self.alpha_t[prev_timestep], self.alpha_t[timestep]
        sigma_t, sigma_s = self.sigma_t[prev_timestep], self.sigma_t[timestep]
        h = lambda_t - lambda_s
        if self.config.algorithm_type == "dpmsolver++":
            x_t = (sigma_t / sigma_s) * sample - (alpha_t * (torch.exp(-h) - 1.0)) * model_output
        elif self.config.algorithm_type == "dpmsolver":
            x_t = (alpha_t / alpha_s) * sample - (sigma_t * (torch.exp(h) - 1.0)) * model_output
        return x_t

    def singlestep_dpm_solver_second_order_update(
        self,
        model_output_list: List[torch.FloatTensor],
        timestep_list: List[int],
        prev_timestep: int,
        sample: torch.FloatTensor,
    ) -> torch.FloatTensor:
        """
        One step for the second-order singlestep DPM-Solver.

        It computes the solution at time `prev_timestep` from the time `timestep_list[-2]`.

        Args:
            model_output_list (`List[torch.FloatTensor]`):
                direct outputs from learned diffusion model at current and latter timesteps.
            timestep (`int`): current and latter discrete timestep in the diffusion chain.
            prev_timestep (`int`): previous discrete timestep in the diffusion chain.
            sample (`torch.FloatTensor`):
                current instance of sample being created by diffusion process.

        Returns:
            `torch.FloatTensor`: the sample tensor at the previous timestep.
        """
        t, s0, s1 = prev_timestep, timestep_list[-1], timestep_list[-2]
        m0, m1 = model_output_list[-1], model_output_list[-2]
        lambda_t, lambda_s0, lambda_s1 = self.lambda_t[t], self.lambda_t[s0], self.lambda_t[s1]
        alpha_t, alpha_s1 = self.alpha_t[t], self.alpha_t[s1]
        sigma_t, sigma_s1 = self.sigma_t[t], self.sigma_t[s1]
        h, h_0 = lambda_t - lambda_s1, lambda_s0 - lambda_s1
        r0 = h_0 / h
        D0, D1 = m1, (1.0 / r0) * (m0 - m1)
        if self.config.algorithm_type == "dpmsolver++":
            # See https://arxiv.org/abs/2211.01095 for detailed derivations
            if self.config.solver_type == "midpoint":
                x_t = (
                    (sigma_t / sigma_s1) * sample
                    - (alpha_t * (torch.exp(-h) - 1.0)) * D0
                    - 0.5 * (alpha_t * (torch.exp(-h) - 1.0)) * D1
                )
            elif self.config.solver_type == "heun":
                x_t = (
                    (sigma_t / sigma_s1) * sample
                    - (alpha_t * (torch.exp(-h) - 1.0)) * D0
                    + (alpha_t * ((torch.exp(-h) - 1.0) / h + 1.0)) * D1
                )
        elif self.config.algorithm_type == "dpmsolver":
            # See https://arxiv.org/abs/2206.00927 for detailed derivations
            if self.config.solver_type == "midpoint":
                x_t = (
                    (alpha_t / alpha_s1) * sample
                    - (sigma_t * (torch.exp(h) - 1.0)) * D0
                    - 0.5 * (sigma_t * (torch.exp(h) - 1.0)) * D1
                )
            elif self.config.solver_type == "heun":
                x_t = (
                    (alpha_t / alpha_s1) * sample
                    - (sigma_t * (torch.exp(h) - 1.0)) * D0
                    - (sigma_t * ((torch.exp(h) - 1.0) / h - 1.0)) * D1
                )
        return x_t

    def singlestep_dpm_solver_third_order_update(
        self,
        model_output_list: List[torch.FloatTensor],
        timestep_list: List[int],
        prev_timestep: int,
        sample: torch.FloatTensor,
    ) -> torch.FloatTensor:
        """
        One step for the third-order singlestep DPM-Solver.

        It computes the solution at time `prev_timestep` from the time `timestep_list[-3]`.

        Args:
            model_output_list (`List[torch.FloatTensor]`):
                direct outputs from learned diffusion model at current and latter timesteps.
            timestep (`int`): current and latter discrete timestep in the diffusion chain.
            prev_timestep (`int`): previous discrete timestep in the diffusion chain.
            sample (`torch.FloatTensor`):
                current instance of sample being created by diffusion process.

        Returns:
            `torch.FloatTensor`: the sample tensor at the previous timestep.
        """
        t, s0, s1, s2 = prev_timestep, timestep_list[-1], timestep_list[-2], timestep_list[-3]
        m0, m1, m2 = model_output_list[-1], model_output_list[-2], model_output_list[-3]
        lambda_t, lambda_s0, lambda_s1, lambda_s2 = (
            self.lambda_t[t],
            self.lambda_t[s0],
            self.lambda_t[s1],
            self.lambda_t[s2],
        )
        alpha_t, alpha_s2 = self.alpha_t[t], self.alpha_t[s2]
        sigma_t, sigma_s2 = self.sigma_t[t], self.sigma_t[s2]
        h, h_0, h_1 = lambda_t - lambda_s2, lambda_s0 - lambda_s2, lambda_s1 - lambda_s2
        r0, r1 = h_0 / h, h_1 / h
        D0 = m2
        D1_0, D1_1 = (1.0 / r1) * (m1 - m2), (1.0 / r0) * (m0 - m2)
        D1 = (r0 * D1_0 - r1 * D1_1) / (r0 - r1)
        D2 = 2.0 * (D1_1 - D1_0) / (r0 - r1)
        if self.config.algorithm_type == "dpmsolver++":
            # See https://arxiv.org/abs/2206.00927 for detailed derivations
            if self.config.solver_type == "midpoint":
                x_t = (
                    (sigma_t / sigma_s2) * sample
                    - (alpha_t * (torch.exp(-h) - 1.0)) * D0
                    + (alpha_t * ((torch.exp(-h) - 1.0) / h + 1.0)) * D1_1
                )
            elif self.config.solver_type == "heun":
                x_t = (
                    (sigma_t / sigma_s2) * sample
                    - (alpha_t * (torch.exp(-h) - 1.0)) * D0
                    + (alpha_t * ((torch.exp(-h) - 1.0) / h + 1.0)) * D1
                    - (alpha_t * ((torch.exp(-h) - 1.0 + h) / h**2 - 0.5)) * D2
                )
        elif self.config.algorithm_type == "dpmsolver":
            # See https://arxiv.org/abs/2206.00927 for detailed derivations
            if self.config.solver_type == "midpoint":
                x_t = (
                    (alpha_t / alpha_s2) * sample
                    - (sigma_t * (torch.exp(h) - 1.0)) * D0
                    - (sigma_t * ((torch.exp(h) - 1.0) / h - 1.0)) * D1_1
                )
            elif self.config.solver_type == "heun":
                x_t = (
                    (alpha_t / alpha_s2) * sample
                    - (sigma_t * (torch.exp(h) - 1.0)) * D0
                    - (sigma_t * ((torch.exp(h) - 1.0) / h - 1.0)) * D1
                    - (sigma_t * ((torch.exp(h) - 1.0 - h) / h**2 - 0.5)) * D2
                )
        return x_t

    def singlestep_dpm_solver_update(
        self,
        model_output_list: List[torch.FloatTensor],
        timestep_list: List[int],
        prev_timestep: int,
        sample: torch.FloatTensor,
        order: int,
    ) -> torch.FloatTensor:
        """
        One step for the singlestep DPM-Solver.

        Args:
            model_output_list (`List[torch.FloatTensor]`):
                direct outputs from learned diffusion model at current and latter timesteps.
            timestep (`int`): current and latter discrete timestep in the diffusion chain.
            prev_timestep (`int`): previous discrete timestep in the diffusion chain.
            sample (`torch.FloatTensor`):
                current instance of sample being created by diffusion process.
            order (`int`):
                the solver order at this step.

        Returns:
            `torch.FloatTensor`: the sample tensor at the previous timestep.
        """
        if order == 1:
            return self.dpm_solver_first_order_update(model_output_list[-1], timestep_list[-1], prev_timestep, sample)
        elif order == 2:
            return self.singlestep_dpm_solver_second_order_update(
                model_output_list, timestep_list, prev_timestep, sample
            )
        elif order == 3:
            return self.singlestep_dpm_solver_third_order_update(
                model_output_list, timestep_list, prev_timestep, sample
            )
        else:
            raise ValueError(f"Order must be 1, 2, 3, got {order}")

    def step(
        self,
        model_output: torch.FloatTensor,
        timestep: int,
        sample: torch.FloatTensor,
        return_dict: bool = True,
    ) -> Union[SchedulerOutput, Tuple]:
        """
        Step function propagating the sample with the singlestep DPM-Solver.

        Args:
            model_output (`torch.FloatTensor`): direct output from learned diffusion model.
            timestep (`int`): current discrete timestep in the diffusion chain.
            sample (`torch.FloatTensor`):
                current instance of sample being created by diffusion process.
            return_dict (`bool`): option for returning tuple rather than SchedulerOutput class

        Returns:
            [`~scheduling_utils.SchedulerOutput`] or `tuple`: [`~scheduling_utils.SchedulerOutput`] if `return_dict` is
            True, otherwise a `tuple`. When returning a tuple, the first element is the sample tensor.

        """
        if self.num_inference_steps is None:
            raise ValueError(
                "Number of inference steps is 'None', you need to run 'set_timesteps' after creating the scheduler"
            )

        if isinstance(timestep, torch.Tensor):
            timestep = timestep.to(self.timesteps.device)
        step_index = (self.timesteps == timestep).nonzero()
        if len(step_index) == 0:
            step_index = len(self.timesteps) - 1
        else:
            step_index = step_index.item()
        prev_timestep = 0 if step_index == len(self.timesteps) - 1 else self.timesteps[step_index + 1]

        model_output = self.convert_model_output(model_output, timestep, sample)
        for i in range(self.config.solver_order - 1):
            self.model_outputs[i] = self.model_outputs[i + 1]
        self.model_outputs[-1] = model_output

        order = self.order_list[step_index]
        # For single-step solvers, we use the initial value at each time with order = 1.
        if order == 1:
            self.sample = sample

        timestep_list = [self.timesteps[step_index - i] for i in range(order - 1, 0, -1)] + [timestep]
        prev_sample = self.singlestep_dpm_solver_update(
            self.model_outputs, timestep_list, prev_timestep, self.sample, order
        )

        if not return_dict:
            return (prev_sample,)

        return SchedulerOutput(prev_sample=prev_sample)

    def scale_model_input(self, sample: torch.FloatTensor, *args, **kwargs) -> torch.FloatTensor:
        """
        Ensures interchangeability with schedulers that need to scale the denoising model input depending on the
        current timestep.

        Args:
            sample (`torch.FloatTensor`): input sample

        Returns:
            `torch.FloatTensor`: scaled input sample
        """
        return sample

    # Copied from diffusers.schedulers.scheduling_ddpm.DDPMScheduler.add_noise
    def add_noise(
        self,
        original_samples: torch.FloatTensor,
        noise: torch.FloatTensor,
        timesteps: torch.IntTensor,
    ) -> torch.FloatTensor:
        # Make sure alphas_cumprod and timestep have same device and dtype as original_samples
        alphas_cumprod = self.alphas_cumprod.to(device=original_samples.device, dtype=original_samples.dtype)
        timesteps = timesteps.to(original_samples.device)

        sqrt_alpha_prod = alphas_cumprod[timesteps] ** 0.5
        sqrt_alpha_prod = sqrt_alpha_prod.flatten()
        while len(sqrt_alpha_prod.shape) < len(original_samples.shape):
            sqrt_alpha_prod = sqrt_alpha_prod.unsqueeze(-1)

        sqrt_one_minus_alpha_prod = (1 - alphas_cumprod[timesteps]) ** 0.5
        sqrt_one_minus_alpha_prod = sqrt_one_minus_alpha_prod.flatten()
        while len(sqrt_one_minus_alpha_prod.shape) < len(original_samples.shape):
            sqrt_one_minus_alpha_prod = sqrt_one_minus_alpha_prod.unsqueeze(-1)

        noisy_samples = sqrt_alpha_prod * original_samples + sqrt_one_minus_alpha_prod * noise
        return noisy_samples

    def __len__(self):
        return self.config.num_train_timesteps