scheduling_dpmsolver_sde.py

# Copyright 2025 Katherine Crowson, The HuggingFace Team and hlky. 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.
# 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.

import math
from dataclasses import dataclass
from typing import List, Literal, Optional, Tuple, Union

import numpy as np
import torch
import torchsde

from ..configuration_utils import ConfigMixin, register_to_config
from ..utils import BaseOutput, is_scipy_available
from .scheduling_utils import KarrasDiffusionSchedulers, SchedulerMixin


if is_scipy_available():
    import scipy.stats


@dataclass
# Copied from diffusers.schedulers.scheduling_ddpm.DDPMSchedulerOutput with DDPM->DPMSolverSDE
class DPMSolverSDESchedulerOutput(BaseOutput):
    """
    Output class for the scheduler's `step` function output.

    Args:
        prev_sample (`torch.Tensor` of shape `(batch_size, num_channels, height, width)` for images):
            Computed sample `(x_{t-1})` of previous timestep. `prev_sample` should be used as next model input in the
            denoising loop.
        pred_original_sample (`torch.Tensor` of shape `(batch_size, num_channels, height, width)` for images):
            The predicted denoised sample `(x_{0})` based on the model output from the current timestep.
            `pred_original_sample` can be used to preview progress or for guidance.
    """

    prev_sample: torch.Tensor
    pred_original_sample: Optional[torch.Tensor] = None


class BatchedBrownianTree:
    """A wrapper around torchsde.BrownianTree that enables batches of entropy."""

    def __init__(self, x, t0, t1, seed=None, **kwargs):
        t0, t1, self.sign = self.sort(t0, t1)
        w0 = kwargs.get("w0", torch.zeros_like(x))
        if seed is None:
            seed = torch.randint(0, 2**63 - 1, []).item()
        self.batched = True
        try:
            assert len(seed) == x.shape[0]
            w0 = w0[0]
        except TypeError:
            seed = [seed]
            self.batched = False
        self.trees = [
            torchsde.BrownianInterval(
                t0=t0,
                t1=t1,
                size=w0.shape,
                dtype=w0.dtype,
                device=w0.device,
                entropy=s,
                tol=1e-6,
                pool_size=24,
                halfway_tree=True,
            )
            for s in seed
        ]

    @staticmethod
    def sort(a, b):
        return (a, b, 1) if a < b else (b, a, -1)

    def __call__(self, t0, t1):
        t0, t1, sign = self.sort(t0, t1)
        w = torch.stack([tree(t0, t1) for tree in self.trees]) * (self.sign * sign)
        return w if self.batched else w[0]


class BrownianTreeNoiseSampler:
    """A noise sampler backed by a torchsde.BrownianTree.

    Args:
        x (Tensor): The tensor whose shape, device and dtype to use to generate
            random samples.
        sigma_min (float): The low end of the valid interval.
        sigma_max (float): The high end of the valid interval.
        seed (int or List[int]): The random seed. If a list of seeds is
            supplied instead of a single integer, then the noise sampler will use one BrownianTree per batch item, each
            with its own seed.
        transform (callable): A function that maps sigma to the sampler's
            internal timestep.
    """

    def __init__(self, x, sigma_min, sigma_max, seed=None, transform=lambda x: x):
        self.transform = transform
        t0, t1 = self.transform(torch.as_tensor(sigma_min)), self.transform(torch.as_tensor(sigma_max))
        self.tree = BatchedBrownianTree(x, t0, t1, seed)

    def __call__(self, sigma, sigma_next):
        t0, t1 = self.transform(torch.as_tensor(sigma)), self.transform(torch.as_tensor(sigma_next))
        return self.tree(t0, t1) / (t1 - t0).abs().sqrt()


# Copied from diffusers.schedulers.scheduling_ddpm.betas_for_alpha_bar
def betas_for_alpha_bar(
    num_diffusion_timesteps: int,
    max_beta: float = 0.999,
    alpha_transform_type: Literal["cosine", "exp"] = "cosine",
) -> torch.Tensor:
    """
    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`, defaults to `0.999`):
            The maximum beta to use; use values lower than 1 to avoid numerical instability.
        alpha_transform_type (`"cosine"` or `"exp"`, defaults to `"cosine"`):
            The type of noise schedule for `alpha_bar`. Choose from `cosine` or `exp`.

    Returns:
        `torch.Tensor`:
            The betas used by the scheduler to step the model outputs.
    """
    if alpha_transform_type == "cosine":

        def alpha_bar_fn(t):
            return math.cos((t + 0.008) / 1.008 * math.pi / 2) ** 2

    elif alpha_transform_type == "exp":

        def alpha_bar_fn(t):
            return math.exp(t * -12.0)

    else:
        raise ValueError(f"Unsupported alpha_transform_type: {alpha_transform_type}")

    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_fn(t2) / alpha_bar_fn(t1), max_beta))
    return torch.tensor(betas, dtype=torch.float32)


class DPMSolverSDEScheduler(SchedulerMixin, ConfigMixin):
    """
    DPMSolverSDEScheduler implements the stochastic sampler from the [Elucidating the Design Space of Diffusion-Based
    Generative Models](https://huggingface.co/papers/2206.00364) paper.

    This model inherits from [`SchedulerMixin`] and [`ConfigMixin`]. Check the superclass documentation for the generic
    methods the library implements for all schedulers such as loading and saving.

    Args:
        num_train_timesteps (`int`, defaults to 1000):
            The number of diffusion steps to train the model.
        beta_start (`float`, defaults to 0.00085):
            The starting `beta` value of inference.
        beta_end (`float`, defaults to 0.012):
            The final `beta` value.
        beta_schedule (`str`, defaults to `"linear"`):
            The beta schedule, a mapping from a beta range to a sequence of betas for stepping the model. Choose from
            `linear` or `scaled_linear`.
        trained_betas (`np.ndarray`, *optional*):
            Pass an array of betas directly to the constructor to bypass `beta_start` and `beta_end`.
        prediction_type (`str`, defaults to `epsilon`, *optional*):
            Prediction type of the scheduler function; can be `epsilon` (predicts the noise of the diffusion process),
            `sample` (directly predicts the noisy sample`) or `v_prediction` (see section 2.4 of [Imagen
            Video](https://huggingface.co/papers/2210.02303) paper).
        use_karras_sigmas (`bool`, *optional*, defaults to `False`):
            Whether to use Karras sigmas for step sizes in the noise schedule during the sampling process. If `True`,
            the sigmas are determined according to a sequence of noise levels {σi}.
        use_exponential_sigmas (`bool`, *optional*, defaults to `False`):
            Whether to use exponential sigmas for step sizes in the noise schedule during the sampling process.
        use_beta_sigmas (`bool`, *optional*, defaults to `False`):
            Whether to use beta sigmas for step sizes in the noise schedule during the sampling process. Refer to [Beta
            Sampling is All You Need](https://huggingface.co/papers/2407.12173) for more information.
        noise_sampler_seed (`int`, *optional*, defaults to `None`):
            The random seed to use for the noise sampler. If `None`, a random seed is generated.
        timestep_spacing (`str`, defaults to `"linspace"`):
            The way the timesteps should be scaled. Refer to Table 2 of the [Common Diffusion Noise Schedules and
            Sample Steps are Flawed](https://huggingface.co/papers/2305.08891) for more information.
        steps_offset (`int`, defaults to 0):
            An offset added to the inference steps, as required by some model families.
    """

    _compatibles = [e.name for e in KarrasDiffusionSchedulers]
    order = 2

    @register_to_config
    def __init__(
        self,
        num_train_timesteps: int = 1000,
        beta_start: float = 0.00085,  # sensible defaults
        beta_end: float = 0.012,
        beta_schedule: str = "linear",
        trained_betas: Optional[Union[np.ndarray, List[float]]] = None,
        prediction_type: str = "epsilon",
        use_karras_sigmas: Optional[bool] = False,
        use_exponential_sigmas: Optional[bool] = False,
        use_beta_sigmas: Optional[bool] = False,
        noise_sampler_seed: Optional[int] = None,
        timestep_spacing: str = "linspace",
        steps_offset: int = 0,
    ):
        if self.config.use_beta_sigmas and not is_scipy_available():
            raise ImportError("Make sure to install scipy if you want to use beta sigmas.")
        if sum([self.config.use_beta_sigmas, self.config.use_exponential_sigmas, self.config.use_karras_sigmas]) > 1:
            raise ValueError(
                "Only one of `config.use_beta_sigmas`, `config.use_exponential_sigmas`, `config.use_karras_sigmas` can be used."
            )
        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} is not implemented for {self.__class__}")

        self.alphas = 1.0 - self.betas
        self.alphas_cumprod = torch.cumprod(self.alphas, dim=0)

        #  set all values
        self.set_timesteps(num_train_timesteps, None, num_train_timesteps)
        self.use_karras_sigmas = use_karras_sigmas
        self.noise_sampler = None
        self.noise_sampler_seed = noise_sampler_seed
        self._step_index = None
        self._begin_index = None
        self.sigmas = self.sigmas.to("cpu")  # to avoid too much CPU/GPU communication

    # Copied from diffusers.schedulers.scheduling_euler_discrete.EulerDiscreteScheduler.index_for_timestep
    def index_for_timestep(
        self, timestep: Union[float, torch.Tensor], schedule_timesteps: Optional[torch.Tensor] = None
    ) -> int:
        """
        Find the index of a given timestep in the timestep schedule.

        Args:
            timestep (`float` or `torch.Tensor`):
                The timestep value to find in the schedule.
            schedule_timesteps (`torch.Tensor`, *optional*):
                The timestep schedule to search in. If `None`, uses `self.timesteps`.

        Returns:
            `int`:
                The index of the timestep in the schedule. For the very first step, returns the second index if
                multiple matches exist to avoid skipping a sigma when starting mid-schedule (e.g., for image-to-image).
        """
        if schedule_timesteps is None:
            schedule_timesteps = self.timesteps

        indices = (schedule_timesteps == timestep).nonzero()

        # The sigma index that is taken for the **very** first `step`
        # is always the second index (or the last index if there is only 1)
        # This way we can ensure we don't accidentally skip a sigma in
        # case we start in the middle of the denoising schedule (e.g. for image-to-image)
        pos = 1 if len(indices) > 1 else 0

        return indices[pos].item()

    # Copied from diffusers.schedulers.scheduling_euler_discrete.EulerDiscreteScheduler._init_step_index
    def _init_step_index(self, timestep: Union[float, torch.Tensor]) -> None:
        """
        Initialize the step index for the scheduler based on the given timestep.

        Args:
            timestep (`float` or `torch.Tensor`):
                The current timestep to initialize the step index from.
        """
        if self.begin_index is None:
            if isinstance(timestep, torch.Tensor):
                timestep = timestep.to(self.timesteps.device)
            self._step_index = self.index_for_timestep(timestep)
        else:
            self._step_index = self._begin_index

    @property
    def init_noise_sigma(self):
        # standard deviation of the initial noise distribution
        if self.config.timestep_spacing in ["linspace", "trailing"]:
            return self.sigmas.max()

        return (self.sigmas.max() ** 2 + 1) ** 0.5

    @property
    def step_index(self):
        """
        The index counter for current timestep. It will increase 1 after each scheduler step.
        """
        return self._step_index

    @property
    def begin_index(self):
        """
        The index for the first timestep. It should be set from pipeline with `set_begin_index` method.
        """
        return self._begin_index

    # Copied from diffusers.schedulers.scheduling_dpmsolver_multistep.DPMSolverMultistepScheduler.set_begin_index
    def set_begin_index(self, begin_index: int = 0):
        """
        Sets the begin index for the scheduler. This function should be run from pipeline before the inference.

        Args:
            begin_index (`int`, defaults to `0`):
                The begin index for the scheduler.
        """
        self._begin_index = begin_index

    def scale_model_input(
        self,
        sample: torch.Tensor,
        timestep: Union[float, torch.Tensor],
    ) -> torch.Tensor:
        """
        Ensures interchangeability with schedulers that need to scale the denoising model input depending on the
        current timestep.

        Args:
            sample (`torch.Tensor`):
                The input sample.
            timestep (`int`, *optional*):
                The current timestep in the diffusion chain.

        Returns:
            `torch.Tensor`:
                A scaled input sample.
        """
        if self.step_index is None:
            self._init_step_index(timestep)

        sigma = self.sigmas[self.step_index]
        sigma_input = sigma if self.state_in_first_order else self.mid_point_sigma
        sample = sample / ((sigma_input**2 + 1) ** 0.5)
        return sample

    def set_timesteps(
        self,
        num_inference_steps: int,
        device: Union[str, torch.device] = None,
        num_train_timesteps: Optional[int] = None,
    ):
        """
        Sets the discrete timesteps used for the diffusion chain (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

        num_train_timesteps = num_train_timesteps or self.config.num_train_timesteps

        # "linspace", "leading", "trailing" corresponds to annotation of Table 2. of https://huggingface.co/papers/2305.08891
        if self.config.timestep_spacing == "linspace":
            timesteps = np.linspace(0, num_train_timesteps - 1, num_inference_steps, dtype=float)[::-1].copy()
        elif self.config.timestep_spacing == "leading":
            step_ratio = num_train_timesteps // self.num_inference_steps
            # creates integer timesteps by multiplying by ratio
            # casting to int to avoid issues when num_inference_step is power of 3
            timesteps = (np.arange(0, num_inference_steps) * step_ratio).round()[::-1].copy().astype(float)
            timesteps += self.config.steps_offset
        elif self.config.timestep_spacing == "trailing":
            step_ratio = num_train_timesteps / self.num_inference_steps
            # creates integer timesteps by multiplying by ratio
            # casting to int to avoid issues when num_inference_step is power of 3
            timesteps = (np.arange(num_train_timesteps, 0, -step_ratio)).round().copy().astype(float)
            timesteps -= 1
        else:
            raise ValueError(
                f"{self.config.timestep_spacing} is not supported. Please make sure to choose one of 'linspace', 'leading' or 'trailing'."
            )

        sigmas = np.array(((1 - self.alphas_cumprod) / self.alphas_cumprod) ** 0.5)
        log_sigmas = np.log(sigmas)
        sigmas = np.interp(timesteps, np.arange(0, len(sigmas)), sigmas)

        if self.config.use_karras_sigmas:
            sigmas = self._convert_to_karras(in_sigmas=sigmas)
            timesteps = np.array([self._sigma_to_t(sigma, log_sigmas) for sigma in sigmas])
        elif self.config.use_exponential_sigmas:
            sigmas = self._convert_to_exponential(in_sigmas=sigmas, num_inference_steps=num_inference_steps)
            timesteps = np.array([self._sigma_to_t(sigma, log_sigmas) for sigma in sigmas])
        elif self.config.use_beta_sigmas:
            sigmas = self._convert_to_beta(in_sigmas=sigmas, num_inference_steps=num_inference_steps)
            timesteps = np.array([self._sigma_to_t(sigma, log_sigmas) for sigma in sigmas])

        second_order_timesteps = self._second_order_timesteps(sigmas, log_sigmas)

        sigmas = np.concatenate([sigmas, [0.0]]).astype(np.float32)
        sigmas = torch.from_numpy(sigmas).to(device=device)
        self.sigmas = torch.cat([sigmas[:1], sigmas[1:-1].repeat_interleave(2), sigmas[-1:]])

        timesteps = torch.from_numpy(timesteps)
        second_order_timesteps = torch.from_numpy(second_order_timesteps)
        timesteps = torch.cat([timesteps[:1], timesteps[1:].repeat_interleave(2)])
        timesteps[1::2] = second_order_timesteps

        if str(device).startswith("mps"):
            # mps does not support float64
            self.timesteps = timesteps.to(device, dtype=torch.float32)
        else:
            self.timesteps = timesteps.to(device=device)

        # empty first order variables
        self.sample = None
        self.mid_point_sigma = None

        self._step_index = None
        self._begin_index = None
        self.sigmas = self.sigmas.to("cpu")  # to avoid too much CPU/GPU communication
        self.noise_sampler = None

    def _second_order_timesteps(self, sigmas, log_sigmas):
        def sigma_fn(_t):
            return np.exp(-_t)

        def t_fn(_sigma):
            return -np.log(_sigma)

        midpoint_ratio = 0.5
        t = t_fn(sigmas)
        delta_time = np.diff(t)
        t_proposed = t[:-1] + delta_time * midpoint_ratio
        sig_proposed = sigma_fn(t_proposed)
        timesteps = np.array([self._sigma_to_t(sigma, log_sigmas) for sigma in sig_proposed])
        return timesteps

    # Copied from diffusers.schedulers.scheduling_euler_discrete.EulerDiscreteScheduler._sigma_to_t
    def _sigma_to_t(self, sigma, log_sigmas):
        """
        Convert sigma values to corresponding timestep values through interpolation.

        Args:
            sigma (`np.ndarray`):
                The sigma value(s) to convert to timestep(s).
            log_sigmas (`np.ndarray`):
                The logarithm of the sigma schedule used for interpolation.

        Returns:
            `np.ndarray`:
                The interpolated timestep value(s) corresponding to the input sigma(s).
        """
        # get log sigma
        log_sigma = np.log(np.maximum(sigma, 1e-10))

        # get distribution
        dists = log_sigma - log_sigmas[:, np.newaxis]

        # get sigmas range
        low_idx = np.cumsum((dists >= 0), axis=0).argmax(axis=0).clip(max=log_sigmas.shape[0] - 2)
        high_idx = low_idx + 1

        low = log_sigmas[low_idx]
        high = log_sigmas[high_idx]

        # interpolate sigmas
        w = (low - log_sigma) / (low - high)
        w = np.clip(w, 0, 1)

        # transform interpolation to time range
        t = (1 - w) * low_idx + w * high_idx
        t = t.reshape(sigma.shape)
        return t

    # Copied from diffusers.schedulers.scheduling_lms_discrete.LMSDiscreteScheduler._convert_to_karras
    def _convert_to_karras(self, in_sigmas: torch.Tensor) -> torch.Tensor:
        """
        Construct the noise schedule as proposed in [Elucidating the Design Space of Diffusion-Based Generative
        Models](https://huggingface.co/papers/2206.00364).

        Args:
            in_sigmas (`torch.Tensor`):
                The input sigma values to be converted.

        Returns:
            `torch.Tensor`:
                The converted sigma values following the Karras noise schedule.
        """

        sigma_min: float = in_sigmas[-1].item()
        sigma_max: float = in_sigmas[0].item()

        rho = 7.0  # 7.0 is the value used in the paper
        ramp = np.linspace(0, 1, self.num_inference_steps)
        min_inv_rho = sigma_min ** (1 / rho)
        max_inv_rho = sigma_max ** (1 / rho)
        sigmas = (max_inv_rho + ramp * (min_inv_rho - max_inv_rho)) ** rho
        return sigmas

    # Copied from diffusers.schedulers.scheduling_euler_discrete.EulerDiscreteScheduler._convert_to_exponential
    def _convert_to_exponential(self, in_sigmas: torch.Tensor, num_inference_steps: int) -> torch.Tensor:
        """
        Construct an exponential noise schedule.

        Args:
            in_sigmas (`torch.Tensor`):
                The input sigma values to be converted.
            num_inference_steps (`int`):
                The number of inference steps to generate the noise schedule for.

        Returns:
            `torch.Tensor`:
                The converted sigma values following an exponential schedule.
        """

        # Hack to make sure that other schedulers which copy this function don't break
        # TODO: Add this logic to the other schedulers
        if hasattr(self.config, "sigma_min"):
            sigma_min = self.config.sigma_min
        else:
            sigma_min = None

        if hasattr(self.config, "sigma_max"):
            sigma_max = self.config.sigma_max
        else:
            sigma_max = None

        sigma_min = sigma_min if sigma_min is not None else in_sigmas[-1].item()
        sigma_max = sigma_max if sigma_max is not None else in_sigmas[0].item()

        sigmas = np.exp(np.linspace(math.log(sigma_max), math.log(sigma_min), num_inference_steps))
        return sigmas

    # Copied from diffusers.schedulers.scheduling_euler_discrete.EulerDiscreteScheduler._convert_to_beta
    def _convert_to_beta(
        self, in_sigmas: torch.Tensor, num_inference_steps: int, alpha: float = 0.6, beta: float = 0.6
    ) -> torch.Tensor:
        """
        Construct a beta noise schedule as proposed in [Beta Sampling is All You
        Need](https://huggingface.co/papers/2407.12173).

        Args:
            in_sigmas (`torch.Tensor`):
                The input sigma values to be converted.
            num_inference_steps (`int`):
                The number of inference steps to generate the noise schedule for.
            alpha (`float`, *optional*, defaults to `0.6`):
                The alpha parameter for the beta distribution.
            beta (`float`, *optional*, defaults to `0.6`):
                The beta parameter for the beta distribution.

        Returns:
            `torch.Tensor`:
                The converted sigma values following a beta distribution schedule.
        """

        # Hack to make sure that other schedulers which copy this function don't break
        # TODO: Add this logic to the other schedulers
        if hasattr(self.config, "sigma_min"):
            sigma_min = self.config.sigma_min
        else:
            sigma_min = None

        if hasattr(self.config, "sigma_max"):
            sigma_max = self.config.sigma_max
        else:
            sigma_max = None

        sigma_min = sigma_min if sigma_min is not None else in_sigmas[-1].item()
        sigma_max = sigma_max if sigma_max is not None else in_sigmas[0].item()

        sigmas = np.array(
            [
                sigma_min + (ppf * (sigma_max - sigma_min))
                for ppf in [
                    scipy.stats.beta.ppf(timestep, alpha, beta)
                    for timestep in 1 - np.linspace(0, 1, num_inference_steps)
                ]
            ]
        )
        return sigmas

    @property
    def state_in_first_order(self):
        return self.sample is None

    def step(
        self,
        model_output: Union[torch.Tensor, np.ndarray],
        timestep: Union[float, torch.Tensor],
        sample: Union[torch.Tensor, np.ndarray],
        return_dict: bool = True,
        s_noise: float = 1.0,
    ) -> Union[DPMSolverSDESchedulerOutput, Tuple]:
        """
        Predict the sample from the previous timestep by reversing the SDE. This function propagates the diffusion
        process from the learned model outputs (most often the predicted noise).

        Args:
            model_output (`torch.Tensor` or `np.ndarray`):
                The direct output from learned diffusion model.
            timestep (`float` or `torch.Tensor`):
                The current discrete timestep in the diffusion chain.
            sample (`torch.Tensor` or `np.ndarray`):
                A current instance of a sample created by the diffusion process.
            return_dict (`bool`):
                Whether or not to return a [`~schedulers.scheduling_dpmsolver_sde.DPMSolverSDESchedulerOutput`] or
                tuple.
            s_noise (`float`, *optional*, defaults to 1.0):
                Scaling factor for noise added to the sample.

        Returns:
            [`~schedulers.scheduling_dpmsolver_sde.DPMSolverSDESchedulerOutput`] or `tuple`:
                If return_dict is `True`, [`~schedulers.scheduling_dpmsolver_sde.DPMSolverSDESchedulerOutput`] is
                returned, otherwise a tuple is returned where the first element is the sample tensor.
        """
        if self.step_index is None:
            self._init_step_index(timestep)

        # Create a noise sampler if it hasn't been created yet
        if self.noise_sampler is None:
            min_sigma, max_sigma = self.sigmas[self.sigmas > 0].min(), self.sigmas.max()
            self.noise_sampler = BrownianTreeNoiseSampler(sample, min_sigma, max_sigma, self.noise_sampler_seed)

        # Define functions to compute sigma and t from each other
        def sigma_fn(_t: torch.Tensor) -> torch.Tensor:
            return _t.neg().exp()

        def t_fn(_sigma: torch.Tensor) -> torch.Tensor:
            return _sigma.log().neg()

        if self.state_in_first_order:
            sigma = self.sigmas[self.step_index]
            sigma_next = self.sigmas[self.step_index + 1]
        else:
            # 2nd order
            sigma = self.sigmas[self.step_index - 1]
            sigma_next = self.sigmas[self.step_index]

        # Set the midpoint and step size for the current step
        midpoint_ratio = 0.5
        t, t_next = t_fn(sigma), t_fn(sigma_next)
        delta_time = t_next - t
        t_proposed = t + delta_time * midpoint_ratio

        # 1. compute predicted original sample (x_0) from sigma-scaled predicted noise
        if self.config.prediction_type == "epsilon":
            sigma_input = sigma if self.state_in_first_order else sigma_fn(t_proposed)
            pred_original_sample = sample - sigma_input * model_output
        elif self.config.prediction_type == "v_prediction":
            sigma_input = sigma if self.state_in_first_order else sigma_fn(t_proposed)
            pred_original_sample = model_output * (-sigma_input / (sigma_input**2 + 1) ** 0.5) + (
                sample / (sigma_input**2 + 1)
            )
        elif self.config.prediction_type == "sample":
            raise NotImplementedError("prediction_type not implemented yet: sample")
        else:
            raise ValueError(
                f"prediction_type given as {self.config.prediction_type} must be one of `epsilon`, or `v_prediction`"
            )

        if sigma_next == 0:
            derivative = (sample - pred_original_sample) / sigma
            dt = sigma_next - sigma
            prev_sample = sample + derivative * dt
        else:
            if self.state_in_first_order:
                t_next = t_proposed
            else:
                sample = self.sample

            sigma_from = sigma_fn(t)
            sigma_to = sigma_fn(t_next)
            sigma_up = min(sigma_to, (sigma_to**2 * (sigma_from**2 - sigma_to**2) / sigma_from**2) ** 0.5)
            sigma_down = (sigma_to**2 - sigma_up**2) ** 0.5
            ancestral_t = t_fn(sigma_down)
            prev_sample = (sigma_fn(ancestral_t) / sigma_fn(t)) * sample - (
                t - ancestral_t
            ).expm1() * pred_original_sample
            prev_sample = prev_sample + self.noise_sampler(sigma_fn(t), sigma_fn(t_next)) * s_noise * sigma_up

            if self.state_in_first_order:
                # store for 2nd order step
                self.sample = sample
                self.mid_point_sigma = sigma_fn(t_next)
            else:
                # free for "first order mode"
                self.sample = None
                self.mid_point_sigma = None

        # upon completion increase step index by one
        self._step_index += 1

        if not return_dict:
            return (
                prev_sample,
                pred_original_sample,
            )

        return DPMSolverSDESchedulerOutput(prev_sample=prev_sample, pred_original_sample=pred_original_sample)

    # Copied from diffusers.schedulers.scheduling_euler_discrete.EulerDiscreteScheduler.add_noise
    def add_noise(
        self,
        original_samples: torch.Tensor,
        noise: torch.Tensor,
        timesteps: torch.Tensor,
    ) -> torch.Tensor:
        """
        Add noise to the original samples according to the noise schedule at the specified timesteps.

        Args:
            original_samples (`torch.Tensor`):
                The original samples to which noise will be added.
            noise (`torch.Tensor`):
                The noise tensor to add to the original samples.
            timesteps (`torch.Tensor`):
                The timesteps at which to add noise, determining the noise level from the schedule.

        Returns:
            `torch.Tensor`:
                The noisy samples with added noise scaled according to the timestep schedule.
        """
        # Make sure sigmas and timesteps have the same device and dtype as original_samples
        sigmas = self.sigmas.to(device=original_samples.device, dtype=original_samples.dtype)
        if original_samples.device.type == "mps" and torch.is_floating_point(timesteps):
            # mps does not support float64
            schedule_timesteps = self.timesteps.to(original_samples.device, dtype=torch.float32)
            timesteps = timesteps.to(original_samples.device, dtype=torch.float32)
        else:
            schedule_timesteps = self.timesteps.to(original_samples.device)
            timesteps = timesteps.to(original_samples.device)

        # self.begin_index is None when scheduler is used for training, or pipeline does not implement set_begin_index
        if self.begin_index is None:
            step_indices = [self.index_for_timestep(t, schedule_timesteps) for t in timesteps]
        elif self.step_index is not None:
            # add_noise is called after first denoising step (for inpainting)
            step_indices = [self.step_index] * timesteps.shape[0]
        else:
            # add noise is called before first denoising step to create initial latent(img2img)
            step_indices = [self.begin_index] * timesteps.shape[0]

        sigma = sigmas[step_indices].flatten()
        while len(sigma.shape) < len(original_samples.shape):
            sigma = sigma.unsqueeze(-1)

        noisy_samples = original_samples + noise * sigma
        return noisy_samples

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