add Sa-Solver (#5975)

* add Sa-Solver



---------
Co-authored-by: Sayak Paul <spsayakpaul@gmail.com>
Co-authored-by: scxue <xueshuchen17@mails.ucas.edu.cn>
Co-authored-by: jschen <chenjunsong4@h-partners.com>
Co-authored-by: Patrick von Platen <patrick.v.platen@gmail.com>
Co-authored-by: yiyixuxu <yixu310@gmail,com>

src/diffusers/__init__.py

@@ -153,6 +153,7 @@ else:
             "LCMScheduler",
             "PNDMScheduler",
             "RePaintScheduler",
+            "SASolverScheduler",
             "SchedulerMixin",
             "ScoreSdeVeScheduler",
             "UnCLIPScheduler",
@@ -530,6 +531,7 @@ if TYPE_CHECKING or DIFFUSERS_SLOW_IMPORT:
             LCMScheduler,
             PNDMScheduler,
             RePaintScheduler,
+            SASolverScheduler,
             SchedulerMixin,
             ScoreSdeVeScheduler,
             UnCLIPScheduler,

src/diffusers/schedulers/__init__.py

@@ -61,6 +61,7 @@ else:
     _import_structure["scheduling_lcm"] = ["LCMScheduler"]
     _import_structure["scheduling_pndm"] = ["PNDMScheduler"]
     _import_structure["scheduling_repaint"] = ["RePaintScheduler"]
+    _import_structure["scheduling_sasolver"] = ["SASolverScheduler"]
     _import_structure["scheduling_sde_ve"] = ["ScoreSdeVeScheduler"]
     _import_structure["scheduling_unclip"] = ["UnCLIPScheduler"]
     _import_structure["scheduling_unipc_multistep"] = ["UniPCMultistepScheduler"]
@@ -152,6 +153,7 @@ if TYPE_CHECKING or DIFFUSERS_SLOW_IMPORT:
         from .scheduling_lcm import LCMScheduler
         from .scheduling_pndm import PNDMScheduler
         from .scheduling_repaint import RePaintScheduler
+        from .scheduling_sasolver import SASolverScheduler
         from .scheduling_sde_ve import ScoreSdeVeScheduler
         from .scheduling_unclip import UnCLIPScheduler
         from .scheduling_unipc_multistep import UniPCMultistepScheduler

src/diffusers/schedulers/scheduling_sasolver.py

+# Copyright 2023 Shuchen Xue, etc. in University of Chinese Academy of Sciences 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.
+# 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.
+
+# DISCLAIMER: check https://arxiv.org/abs/2309.05019
+# The codebase is modified based on https://github.com/huggingface/diffusers/blob/main/src/diffusers/schedulers/scheduling_dpmsolver_multistep.py
+
+import math
+from typing import Callable, List, Optional, Tuple, Union
+
+import numpy as np
+import torch
+
+from ..configuration_utils import ConfigMixin, register_to_config
+from ..utils import deprecate
+from ..utils.torch_utils import randn_tensor
+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,
+    alpha_transform_type="cosine",
+):
+    """
+    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.
+        alpha_transform_type (`str`, *optional*, default to `cosine`): the type of noise schedule for alpha_bar.
+                     Choose from `cosine` or `exp`
+
+    Returns:
+        betas (`np.ndarray`): 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_tranform_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 SASolverScheduler(SchedulerMixin, ConfigMixin):
+    """
+    `SASolverScheduler` is a fast dedicated high-order solver for diffusion SDEs.
+
+    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 0.0001):
+            The starting `beta` value of inference.
+        beta_end (`float`, defaults to 0.02):
+            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`, `scaled_linear`, or `squaredcos_cap_v2`.
+        trained_betas (`np.ndarray`, *optional*):
+            Pass an array of betas directly to the constructor to bypass `beta_start` and `beta_end`.
+        predictor_order (`int`, defaults to 2):
+            The predictor order which can be `1` or `2` or `3` or '4'. It is recommended to use `predictor_order=2` for guided
+            sampling, and `predictor_order=3` for unconditional sampling.
+        corrector_order (`int`, defaults to 2):
+            The corrector order which can be `1` or `2` or `3` or '4'. It is recommended to use `corrector_order=2` for guided
+            sampling, and `corrector_order=3` for unconditional sampling.
+        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://imagen.research.google/video/paper.pdf) paper).
+        tau_func (`Callable`, *optional*):
+            Stochasticity during the sampling. Default in init is `lambda t: 1 if t >= 200 and t <= 800 else 0`. SA-Solver
+            will sample from vanilla diffusion ODE if tau_func is set to `lambda t: 0`. SA-Solver will sample from vanilla
+            diffusion SDE if tau_func is set to `lambda t: 1`. For more details, please check https://arxiv.org/abs/2309.05019
+        thresholding (`bool`, defaults to `False`):
+            Whether to use the "dynamic thresholding" method. This is unsuitable for latent-space diffusion models such
+            as Stable Diffusion.
+        dynamic_thresholding_ratio (`float`, defaults to 0.995):
+            The ratio for the dynamic thresholding method. Valid only when `thresholding=True`.
+        sample_max_value (`float`, defaults to 1.0):
+            The threshold value for dynamic thresholding. Valid only when `thresholding=True` and
+            `algorithm_type="dpmsolver++"`.
+        algorithm_type (`str`, defaults to `data_prediction`):
+            Algorithm type for the solver; can be `data_prediction` or `noise_prediction`. It is recommended to use `data_prediction`
+            with `solver_order=2` for guided sampling like in Stable Diffusion.
+        lower_order_final (`bool`, defaults to `True`):
+            Whether to use lower-order solvers in the final steps. Default = True.
+        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}.
+        lambda_min_clipped (`float`, defaults to `-inf`):
+            Clipping threshold for the minimum value of `lambda(t)` for numerical stability. This is critical for the
+            cosine (`squaredcos_cap_v2`) noise schedule.
+        variance_type (`str`, *optional*):
+            Set to "learned" or "learned_range" for diffusion models that predict variance. If set, the model's output
+            contains the predicted Gaussian variance.
+        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. You can use a combination of `offset=1` and
+            `set_alpha_to_one=False` to make the last step use step 0 for the previous alpha product like in Stable
+            Diffusion.
+    """
+
+    _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[Union[np.ndarray, List[float]]] = None,
+        predictor_order: int = 2,
+        corrector_order: int = 2,
+        prediction_type: str = "epsilon",
+        tau_func: Optional[Callable] = None,
+        thresholding: bool = False,
+        dynamic_thresholding_ratio: float = 0.995,
+        sample_max_value: float = 1.0,
+        algorithm_type: str = "data_prediction",
+        lower_order_final: bool = True,
+        use_karras_sigmas: Optional[bool] = False,
+        lambda_min_clipped: float = -float("inf"),
+        variance_type: Optional[str] = None,
+        timestep_spacing: str = "linspace",
+        steps_offset: int = 0,
+    ):
+        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)
+        self.sigmas = ((1 - self.alphas_cumprod) / self.alphas_cumprod) ** 0.5
+
+        # standard deviation of the initial noise distribution
+        self.init_noise_sigma = 1.0
+
+        if algorithm_type not in ["data_prediction", "noise_prediction"]:
+            raise NotImplementedError(f"{algorithm_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.timestep_list = [None] * max(predictor_order, corrector_order - 1)
+        self.model_outputs = [None] * max(predictor_order, corrector_order - 1)
+
+        if tau_func is None:
+            self.tau_func = lambda t: 1 if t >= 200 and t <= 800 else 0
+        else:
+            self.tau_func = tau_func
+        self.predict_x0 = algorithm_type == "data_prediction"
+        self.lower_order_nums = 0
+        self.last_sample = None
+        self._step_index = None
+        self.sigmas.to("cpu")  # to avoid too much CPU/GPU communication
+
+    @property
+    def step_index(self):
+        """
+        The index counter for current timestep. It will increae 1 after each scheduler step.
+        """
+        return self._step_index
+
+    def set_timesteps(self, num_inference_steps: int = None, device: Union[str, torch.device] = 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.
+        """
+        # 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)
+        last_timestep = ((self.config.num_train_timesteps - clipped_idx).numpy()).item()
+
+        # "linspace", "leading", "trailing" corresponds to annotation of Table 2. of https://arxiv.org/abs/2305.08891
+        if self.config.timestep_spacing == "linspace":
+            timesteps = (
+                np.linspace(0, last_timestep - 1, num_inference_steps + 1).round()[::-1][:-1].copy().astype(np.int64)
+            )
+
+        elif self.config.timestep_spacing == "leading":
+            step_ratio = last_timestep // (num_inference_steps + 1)
+            # 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 + 1) * step_ratio).round()[::-1][:-1].copy().astype(np.int64)
+            timesteps += self.config.steps_offset
+        elif self.config.timestep_spacing == "trailing":
+            step_ratio = self.config.num_train_timesteps / 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(last_timestep, 0, -step_ratio).round().copy().astype(np.int64)
+            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)
+        if self.config.use_karras_sigmas:
+            log_sigmas = np.log(sigmas)
+            sigmas = np.flip(sigmas).copy()
+            sigmas = self._convert_to_karras(in_sigmas=sigmas, num_inference_steps=num_inference_steps)
+            timesteps = np.array([self._sigma_to_t(sigma, log_sigmas) for sigma in sigmas]).round()
+            sigmas = np.concatenate([sigmas, sigmas[-1:]]).astype(np.float32)
+        else:
+            sigmas = np.interp(timesteps, np.arange(0, len(sigmas)), sigmas)
+            sigma_last = ((1 - self.alphas_cumprod[0]) / self.alphas_cumprod[0]) ** 0.5
+            sigmas = np.concatenate([sigmas, [sigma_last]]).astype(np.float32)
+
+        self.sigmas = torch.from_numpy(sigmas)
+        self.timesteps = torch.from_numpy(timesteps).to(device=device, dtype=torch.int64)
+
+        self.num_inference_steps = len(timesteps)
+        self.model_outputs = [
+            None,
+        ] * max(self.config.predictor_order, self.config.corrector_order - 1)
+        self.lower_order_nums = 0
+        self.last_sample = None
+
+        # add an index counter for schedulers that allow duplicated timesteps
+        self._step_index = None
+        self.sigmas.to("cpu")  # to avoid too much CPU/GPU communication
+
+    # 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, *remaining_dims = 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 * np.prod(remaining_dims))
+
+        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, *remaining_dims)
+        sample = sample.to(dtype)
+
+        return sample
+
+    # Copied from diffusers.schedulers.scheduling_euler_discrete.EulerDiscreteScheduler._sigma_to_t
+    def _sigma_to_t(self, sigma, log_sigmas):
+        # 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_dpmsolver_multistep.DPMSolverMultistepScheduler._sigma_to_alpha_sigma_t
+    def _sigma_to_alpha_sigma_t(self, sigma):
+        alpha_t = 1 / ((sigma**2 + 1) ** 0.5)
+        sigma_t = sigma * alpha_t
+
+        return alpha_t, sigma_t
+
+    # Copied from diffusers.schedulers.scheduling_euler_discrete.EulerDiscreteScheduler._convert_to_karras
+    def _convert_to_karras(self, in_sigmas: torch.FloatTensor, num_inference_steps) -> torch.FloatTensor:
+        """Constructs the noise schedule of Karras et al. (2022)."""
+
+        # 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()
+
+        rho = 7.0  # 7.0 is the value used in the paper
+        ramp = np.linspace(0, 1, 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
+
+    def convert_model_output(
+        self,
+        model_output: torch.FloatTensor,
+        *args,
+        sample: torch.FloatTensor = None,
+        **kwargs,
+    ) -> torch.FloatTensor:
+        """
+        Convert the model output to the corresponding type the data_prediction/noise_prediction algorithm needs. Noise_prediction is
+        designed to discretize an integral of the noise prediction model, and data_prediction is designed to discretize an
+        integral of the data prediction model.
+
+        <Tip>
+
+        The algorithm and model type are decoupled. You can use either data_prediction or noise_prediction for both noise
+        prediction and data prediction models.
+
+        </Tip>
+
+        Args:
+            model_output (`torch.FloatTensor`):
+                The direct output from the learned diffusion model.
+            sample (`torch.FloatTensor`):
+                A current instance of a sample created by the diffusion process.
+
+        Returns:
+            `torch.FloatTensor`:
+                The converted model output.
+        """
+        timestep = args[0] if len(args) > 0 else kwargs.pop("timestep", None)
+        if sample is None:
+            if len(args) > 1:
+                sample = args[1]
+            else:
+                raise ValueError("missing `sample` as a required keyward argument")
+        if timestep is not None:
+            deprecate(
+                "timesteps",
+                "1.0.0",
+                "Passing `timesteps` is deprecated and has no effect as model output conversion is now handled via an internal counter `self.step_index`",
+            )
+
+        sigma = self.sigmas[self.step_index]
+        alpha_t, sigma_t = self._sigma_to_alpha_sigma_t(sigma)
+        # SA-Solver_data_prediction needs to solve an integral of the data prediction model.
+        if self.config.algorithm_type in ["data_prediction"]:
+            if self.config.prediction_type == "epsilon":
+                # SA-Solver only needs the "mean" output.
+                if self.config.variance_type in ["learned", "learned_range"]:
+                    model_output = model_output[:, :3]
+                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":
+                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 SASolverScheduler."
+                )
+
+            if self.config.thresholding:
+                x0_pred = self._threshold_sample(x0_pred)
+
+            return x0_pred
+
+        # SA-Solver_noise_prediction needs to solve an integral of the noise prediction model.
+        elif self.config.algorithm_type in ["noise_prediction"]:
+            if self.config.prediction_type == "epsilon":
+                # SA-Solver only needs the "mean" output.
+                if self.config.variance_type in ["learned", "learned_range"]:
+                    epsilon = model_output[:, :3]
+                else:
+                    epsilon = model_output
+            elif self.config.prediction_type == "sample":
+                epsilon = (sample - alpha_t * model_output) / sigma_t
+            elif self.config.prediction_type == "v_prediction":
+                epsilon = alpha_t * model_output + sigma_t * sample
+            else:
+                raise ValueError(
+                    f"prediction_type given as {self.config.prediction_type} must be one of `epsilon`, `sample`, or"
+                    " `v_prediction` for the SASolverScheduler."
+                )
+
+            if self.config.thresholding:
+                alpha_t, sigma_t = self.alpha_t[timestep], self.sigma_t[timestep]
+                x0_pred = (sample - sigma_t * epsilon) / alpha_t
+                x0_pred = self._threshold_sample(x0_pred)
+                epsilon = (sample - alpha_t * x0_pred) / sigma_t
+
+            return epsilon
+
+    def get_coefficients_exponential_negative(self, order, interval_start, interval_end):
+        """
+        Calculate the integral of exp(-x) * x^order dx from interval_start to interval_end
+        """
+        assert order in [0, 1, 2, 3], "order is only supported for 0, 1, 2 and 3"
+
+        if order == 0:
+            return torch.exp(-interval_end) * (torch.exp(interval_end - interval_start) - 1)
+        elif order == 1:
+            return torch.exp(-interval_end) * (
+                (interval_start + 1) * torch.exp(interval_end - interval_start) - (interval_end + 1)
+            )
+        elif order == 2:
+            return torch.exp(-interval_end) * (
+                (interval_start**2 + 2 * interval_start + 2) * torch.exp(interval_end - interval_start)
+                - (interval_end**2 + 2 * interval_end + 2)
+            )
+        elif order == 3:
+            return torch.exp(-interval_end) * (
+                (interval_start**3 + 3 * interval_start**2 + 6 * interval_start + 6)
+                * torch.exp(interval_end - interval_start)
+                - (interval_end**3 + 3 * interval_end**2 + 6 * interval_end + 6)
+            )
+
+    def get_coefficients_exponential_positive(self, order, interval_start, interval_end, tau):
+        """
+        Calculate the integral of exp(x(1+tau^2)) * x^order dx from interval_start to interval_end
+        """
+        assert order in [0, 1, 2, 3], "order is only supported for 0, 1, 2 and 3"
+
+        # after change of variable(cov)
+        interval_end_cov = (1 + tau**2) * interval_end
+        interval_start_cov = (1 + tau**2) * interval_start
+
+        if order == 0:
+            return (
+                torch.exp(interval_end_cov) * (1 - torch.exp(-(interval_end_cov - interval_start_cov))) / (1 + tau**2)
+            )
+        elif order == 1:
+            return (
+                torch.exp(interval_end_cov)
+                * (
+                    (interval_end_cov - 1)
+                    - (interval_start_cov - 1) * torch.exp(-(interval_end_cov - interval_start_cov))
+                )
+                / ((1 + tau**2) ** 2)
+            )
+        elif order == 2:
+            return (
+                torch.exp(interval_end_cov)
+                * (
+                    (interval_end_cov**2 - 2 * interval_end_cov + 2)
+                    - (interval_start_cov**2 - 2 * interval_start_cov + 2)
+                    * torch.exp(-(interval_end_cov - interval_start_cov))
+                )
+                / ((1 + tau**2) ** 3)
+            )
+        elif order == 3:
+            return (
+                torch.exp(interval_end_cov)
+                * (
+                    (interval_end_cov**3 - 3 * interval_end_cov**2 + 6 * interval_end_cov - 6)
+                    - (interval_start_cov**3 - 3 * interval_start_cov**2 + 6 * interval_start_cov - 6)
+                    * torch.exp(-(interval_end_cov - interval_start_cov))
+                )
+                / ((1 + tau**2) ** 4)
+            )
+
+    def lagrange_polynomial_coefficient(self, order, lambda_list):
+        """
+        Calculate the coefficient of lagrange polynomial
+        """
+
+        assert order in [0, 1, 2, 3]
+        assert order == len(lambda_list) - 1
+        if order == 0:
+            return [[1]]
+        elif order == 1:
+            return [
+                [
+                    1 / (lambda_list[0] - lambda_list[1]),
+                    -lambda_list[1] / (lambda_list[0] - lambda_list[1]),
+                ],
+                [
+                    1 / (lambda_list[1] - lambda_list[0]),
+                    -lambda_list[0] / (lambda_list[1] - lambda_list[0]),
+                ],
+            ]
+        elif order == 2:
+            denominator1 = (lambda_list[0] - lambda_list[1]) * (lambda_list[0] - lambda_list[2])
+            denominator2 = (lambda_list[1] - lambda_list[0]) * (lambda_list[1] - lambda_list[2])
+            denominator3 = (lambda_list[2] - lambda_list[0]) * (lambda_list[2] - lambda_list[1])
+            return [
+                [
+                    1 / denominator1,
+                    (-lambda_list[1] - lambda_list[2]) / denominator1,
+                    lambda_list[1] * lambda_list[2] / denominator1,
+                ],
+                [
+                    1 / denominator2,
+                    (-lambda_list[0] - lambda_list[2]) / denominator2,
+                    lambda_list[0] * lambda_list[2] / denominator2,
+                ],
+                [
+                    1 / denominator3,
+                    (-lambda_list[0] - lambda_list[1]) / denominator3,
+                    lambda_list[0] * lambda_list[1] / denominator3,
+                ],
+            ]
+        elif order == 3:
+            denominator1 = (
+                (lambda_list[0] - lambda_list[1])
+                * (lambda_list[0] - lambda_list[2])
+                * (lambda_list[0] - lambda_list[3])
+            )
+            denominator2 = (
+                (lambda_list[1] - lambda_list[0])
+                * (lambda_list[1] - lambda_list[2])
+                * (lambda_list[1] - lambda_list[3])
+            )
+            denominator3 = (
+                (lambda_list[2] - lambda_list[0])
+                * (lambda_list[2] - lambda_list[1])
+                * (lambda_list[2] - lambda_list[3])
+            )
+            denominator4 = (
+                (lambda_list[3] - lambda_list[0])
+                * (lambda_list[3] - lambda_list[1])
+                * (lambda_list[3] - lambda_list[2])
+            )
+            return [
+                [
+                    1 / denominator1,
+                    (-lambda_list[1] - lambda_list[2] - lambda_list[3]) / denominator1,
+                    (
+                        lambda_list[1] * lambda_list[2]
+                        + lambda_list[1] * lambda_list[3]
+                        + lambda_list[2] * lambda_list[3]
+                    )
+                    / denominator1,
+                    (-lambda_list[1] * lambda_list[2] * lambda_list[3]) / denominator1,
+                ],
+                [
+                    1 / denominator2,
+                    (-lambda_list[0] - lambda_list[2] - lambda_list[3]) / denominator2,
+                    (
+                        lambda_list[0] * lambda_list[2]
+                        + lambda_list[0] * lambda_list[3]
+                        + lambda_list[2] * lambda_list[3]
+                    )
+                    / denominator2,
+                    (-lambda_list[0] * lambda_list[2] * lambda_list[3]) / denominator2,
+                ],
+                [
+                    1 / denominator3,
+                    (-lambda_list[0] - lambda_list[1] - lambda_list[3]) / denominator3,
+                    (
+                        lambda_list[0] * lambda_list[1]
+                        + lambda_list[0] * lambda_list[3]
+                        + lambda_list[1] * lambda_list[3]
+                    )
+                    / denominator3,
+                    (-lambda_list[0] * lambda_list[1] * lambda_list[3]) / denominator3,
+                ],
+                [
+                    1 / denominator4,
+                    (-lambda_list[0] - lambda_list[1] - lambda_list[2]) / denominator4,
+                    (
+                        lambda_list[0] * lambda_list[1]
+                        + lambda_list[0] * lambda_list[2]
+                        + lambda_list[1] * lambda_list[2]
+                    )
+                    / denominator4,
+                    (-lambda_list[0] * lambda_list[1] * lambda_list[2]) / denominator4,
+                ],
+            ]
+
+    def get_coefficients_fn(self, order, interval_start, interval_end, lambda_list, tau):
+        assert order in [1, 2, 3, 4]
+        assert order == len(lambda_list), "the length of lambda list must be equal to the order"
+        coefficients = []
+        lagrange_coefficient = self.lagrange_polynomial_coefficient(order - 1, lambda_list)
+        for i in range(order):
+            coefficient = 0
+            for j in range(order):
+                if self.predict_x0:
+                    coefficient += lagrange_coefficient[i][j] * self.get_coefficients_exponential_positive(
+                        order - 1 - j, interval_start, interval_end, tau
+                    )
+                else:
+                    coefficient += lagrange_coefficient[i][j] * self.get_coefficients_exponential_negative(
+                        order - 1 - j, interval_start, interval_end
+                    )
+            coefficients.append(coefficient)
+        assert len(coefficients) == order, "the length of coefficients does not match the order"
+        return coefficients
+
+    def stochastic_adams_bashforth_update(
+        self,
+        model_output: torch.FloatTensor,
+        *args,
+        sample: torch.FloatTensor,
+        noise: torch.FloatTensor,
+        order: int,
+        tau: torch.FloatTensor,
+        **kwargs,
+    ) -> torch.FloatTensor:
+        """
+        One step for the SA-Predictor.
+
+        Args:
+            model_output (`torch.FloatTensor`):
+                The direct output from the learned diffusion model at the current timestep.
+            prev_timestep (`int`):
+                The previous discrete timestep in the diffusion chain.
+            sample (`torch.FloatTensor`):
+                A current instance of a sample created by the diffusion process.
+            order (`int`):
+                The order of SA-Predictor at this timestep.
+
+        Returns:
+            `torch.FloatTensor`:
+                The sample tensor at the previous timestep.
+        """
+        prev_timestep = args[0] if len(args) > 0 else kwargs.pop("prev_timestep", None)
+        if sample is None:
+            if len(args) > 1:
+                sample = args[1]
+            else:
+                raise ValueError(" missing `sample` as a required keyward argument")
+        if noise is None:
+            if len(args) > 2:
+                noise = args[2]
+            else:
+                raise ValueError(" missing `noise` as a required keyward argument")
+        if order is None:
+            if len(args) > 3:
+                order = args[3]
+            else:
+                raise ValueError(" missing `order` as a required keyward argument")
+        if tau is None:
+            if len(args) > 4:
+                tau = args[4]
+            else:
+                raise ValueError(" missing `tau` as a required keyward argument")
+        if prev_timestep is not None:
+            deprecate(
+                "prev_timestep",
+                "1.0.0",
+                "Passing `prev_timestep` is deprecated and has no effect as model output conversion is now handled via an internal counter `self.step_index`",
+            )
+        model_output_list = self.model_outputs
+        sigma_t, sigma_s0 = (
+            self.sigmas[self.step_index + 1],
+            self.sigmas[self.step_index],
+        )
+        alpha_t, sigma_t = self._sigma_to_alpha_sigma_t(sigma_t)
+        alpha_s0, sigma_s0 = self._sigma_to_alpha_sigma_t(sigma_s0)
+        lambda_t = torch.log(alpha_t) - torch.log(sigma_t)
+        lambda_s0 = torch.log(alpha_s0) - torch.log(sigma_s0)
+
+        gradient_part = torch.zeros_like(sample)
+        h = lambda_t - lambda_s0
+        lambda_list = []
+
+        for i in range(order):
+            si = self.step_index - i
+            alpha_si, sigma_si = self._sigma_to_alpha_sigma_t(self.sigmas[si])
+            lambda_si = torch.log(alpha_si) - torch.log(sigma_si)
+            lambda_list.append(lambda_si)
+
+        gradient_coefficients = self.get_coefficients_fn(order, lambda_s0, lambda_t, lambda_list, tau)
+
+        x = sample
+
+        if self.predict_x0:
+            if (
+                order == 2
+            ):  ## if order = 2 we do a modification that does not influence the convergence order similar to unipc. Note: This is used only for few steps sampling.
+                # The added term is O(h^3). Empirically we find it will slightly improve the image quality.
+                # ODE case
+                # gradient_coefficients[0] += 1.0 * torch.exp(lambda_t) * (h ** 2 / 2 - (h - 1 + torch.exp(-h))) / (ns.marginal_lambda(t_prev_list[-1]) - ns.marginal_lambda(t_prev_list[-2]))
+                # gradient_coefficients[1] -= 1.0 * torch.exp(lambda_t) * (h ** 2 / 2 - (h - 1 + torch.exp(-h))) / (ns.marginal_lambda(t_prev_list[-1]) - ns.marginal_lambda(t_prev_list[-2]))
+                temp_sigma = self.sigmas[self.step_index - 1]
+                temp_alpha_s, temp_sigma_s = self._sigma_to_alpha_sigma_t(temp_sigma)
+                temp_lambda_s = torch.log(temp_alpha_s) - torch.log(temp_sigma_s)
+                gradient_coefficients[0] += (
+                    1.0
+                    * torch.exp((1 + tau**2) * lambda_t)
+                    * (h**2 / 2 - (h * (1 + tau**2) - 1 + torch.exp((1 + tau**2) * (-h))) / ((1 + tau**2) ** 2))
+                    / (lambda_s0 - temp_lambda_s)
+                )
+                gradient_coefficients[1] -= (
+                    1.0
+                    * torch.exp((1 + tau**2) * lambda_t)
+                    * (h**2 / 2 - (h * (1 + tau**2) - 1 + torch.exp((1 + tau**2) * (-h))) / ((1 + tau**2) ** 2))
+                    / (lambda_s0 - temp_lambda_s)
+                )
+
+        for i in range(order):
+            if self.predict_x0:
+                gradient_part += (
+                    (1 + tau**2)
+                    * sigma_t
+                    * torch.exp(-(tau**2) * lambda_t)
+                    * gradient_coefficients[i]
+                    * model_output_list[-(i + 1)]
+                )
+            else:
+                gradient_part += -(1 + tau**2) * alpha_t * gradient_coefficients[i] * model_output_list[-(i + 1)]
+
+        if self.predict_x0:
+            noise_part = sigma_t * torch.sqrt(1 - torch.exp(-2 * tau**2 * h)) * noise
+        else:
+            noise_part = tau * sigma_t * torch.sqrt(torch.exp(2 * h) - 1) * noise
+
+        if self.predict_x0:
+            x_t = torch.exp(-(tau**2) * h) * (sigma_t / sigma_s0) * x + gradient_part + noise_part
+        else:
+            x_t = (alpha_t / alpha_s0) * x + gradient_part + noise_part
+
+        x_t = x_t.to(x.dtype)
+        return x_t
+
+    def stochastic_adams_moulton_update(
+        self,
+        this_model_output: torch.FloatTensor,
+        *args,
+        last_sample: torch.FloatTensor,
+        last_noise: torch.FloatTensor,
+        this_sample: torch.FloatTensor,
+        order: int,
+        tau: torch.FloatTensor,
+        **kwargs,
+    ) -> torch.FloatTensor:
+        """
+        One step for the SA-Corrector.
+
+        Args:
+            this_model_output (`torch.FloatTensor`):
+                The model outputs at `x_t`.
+            this_timestep (`int`):
+                The current timestep `t`.
+            last_sample (`torch.FloatTensor`):
+                The generated sample before the last predictor `x_{t-1}`.
+            this_sample (`torch.FloatTensor`):
+                The generated sample after the last predictor `x_{t}`.
+            order (`int`):
+                The order of SA-Corrector at this step.
+
+        Returns:
+            `torch.FloatTensor`:
+                The corrected sample tensor at the current timestep.
+        """
+
+        this_timestep = args[0] if len(args) > 0 else kwargs.pop("this_timestep", None)
+        if last_sample is None:
+            if len(args) > 1:
+                last_sample = args[1]
+            else:
+                raise ValueError(" missing`last_sample` as a required keyward argument")
+        if last_noise is None:
+            if len(args) > 2:
+                last_noise = args[2]
+            else:
+                raise ValueError(" missing`last_noise` as a required keyward argument")
+        if this_sample is None:
+            if len(args) > 3:
+                this_sample = args[3]
+            else:
+                raise ValueError(" missing`this_sample` as a required keyward argument")
+        if order is None:
+            if len(args) > 4:
+                order = args[4]
+            else:
+                raise ValueError(" missing`order` as a required keyward argument")
+        if tau is None:
+            if len(args) > 5:
+                tau = args[5]
+            else:
+                raise ValueError(" missing`tau` as a required keyward argument")
+        if this_timestep is not None:
+            deprecate(
+                "this_timestep",
+                "1.0.0",
+                "Passing `this_timestep` is deprecated and has no effect as model output conversion is now handled via an internal counter `self.step_index`",
+            )
+
+        model_output_list = self.model_outputs
+        sigma_t, sigma_s0 = (
+            self.sigmas[self.step_index],
+            self.sigmas[self.step_index - 1],
+        )
+        alpha_t, sigma_t = self._sigma_to_alpha_sigma_t(sigma_t)
+        alpha_s0, sigma_s0 = self._sigma_to_alpha_sigma_t(sigma_s0)
+
+        lambda_t = torch.log(alpha_t) - torch.log(sigma_t)
+        lambda_s0 = torch.log(alpha_s0) - torch.log(sigma_s0)
+        gradient_part = torch.zeros_like(this_sample)
+        h = lambda_t - lambda_s0
+        lambda_list = []
+        for i in range(order):
+            si = self.step_index - i
+            alpha_si, sigma_si = self._sigma_to_alpha_sigma_t(self.sigmas[si])
+            lambda_si = torch.log(alpha_si) - torch.log(sigma_si)
+            lambda_list.append(lambda_si)
+
+        model_prev_list = model_output_list + [this_model_output]
+
+        gradient_coefficients = self.get_coefficients_fn(order, lambda_s0, lambda_t, lambda_list, tau)
+
+        x = last_sample
+
+        if self.predict_x0:
+            if (
+                order == 2
+            ):  ## if order = 2 we do a modification that does not influence the convergence order similar to UniPC. Note: This is used only for few steps sampling.
+                # The added term is O(h^3). Empirically we find it will slightly improve the image quality.
+                # ODE case
+                # gradient_coefficients[0] += 1.0 * torch.exp(lambda_t) * (h / 2 - (h - 1 + torch.exp(-h)) / h)
+                # gradient_coefficients[1] -= 1.0 * torch.exp(lambda_t) * (h / 2 - (h - 1 + torch.exp(-h)) / h)
+                gradient_coefficients[0] += (
+                    1.0
+                    * torch.exp((1 + tau**2) * lambda_t)
+                    * (h / 2 - (h * (1 + tau**2) - 1 + torch.exp((1 + tau**2) * (-h))) / ((1 + tau**2) ** 2 * h))
+                )
+                gradient_coefficients[1] -= (
+                    1.0
+                    * torch.exp((1 + tau**2) * lambda_t)
+                    * (h / 2 - (h * (1 + tau**2) - 1 + torch.exp((1 + tau**2) * (-h))) / ((1 + tau**2) ** 2 * h))
+                )
+
+        for i in range(order):
+            if self.predict_x0:
+                gradient_part += (
+                    (1 + tau**2)
+                    * sigma_t
+                    * torch.exp(-(tau**2) * lambda_t)
+                    * gradient_coefficients[i]
+                    * model_prev_list[-(i + 1)]
+                )
+            else:
+                gradient_part += -(1 + tau**2) * alpha_t * gradient_coefficients[i] * model_prev_list[-(i + 1)]
+
+        if self.predict_x0:
+            noise_part = sigma_t * torch.sqrt(1 - torch.exp(-2 * tau**2 * h)) * last_noise
+        else:
+            noise_part = tau * sigma_t * torch.sqrt(torch.exp(2 * h) - 1) * last_noise
+
+        if self.predict_x0:
+            x_t = torch.exp(-(tau**2) * h) * (sigma_t / sigma_s0) * x + gradient_part + noise_part
+        else:
+            x_t = (alpha_t / alpha_s0) * x + gradient_part + noise_part
+
+        x_t = x_t.to(x.dtype)
+        return x_t
+
+    def _init_step_index(self, timestep):
+        if isinstance(timestep, torch.Tensor):
+            timestep = timestep.to(self.timesteps.device)
+
+        index_candidates = (self.timesteps == timestep).nonzero()
+
+        if len(index_candidates) == 0:
+            step_index = len(self.timesteps) - 1
+        # 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)
+        elif len(index_candidates) > 1:
+            step_index = index_candidates[1].item()
+        else:
+            step_index = index_candidates[0].item()
+
+        self._step_index = step_index
+
+    def step(
+        self,
+        model_output: torch.FloatTensor,
+        timestep: int,
+        sample: torch.FloatTensor,
+        generator=None,
+        return_dict: bool = True,
+    ) -> Union[SchedulerOutput, Tuple]:
+        """
+        Predict the sample from the previous timestep by reversing the SDE. This function propagates the sample with
+        the SA-Solver.
+
+        Args:
+            model_output (`torch.FloatTensor`):
+                The direct output from learned diffusion model.
+            timestep (`int`):
+                The current discrete timestep in the diffusion chain.
+            sample (`torch.FloatTensor`):
+                A current instance of a sample created by the diffusion process.
+            generator (`torch.Generator`, *optional*):
+                A random number generator.
+            return_dict (`bool`):
+                Whether or not to return a [`~schedulers.scheduling_utils.SchedulerOutput`] or `tuple`.
+
+        Returns:
+            [`~schedulers.scheduling_utils.SchedulerOutput`] or `tuple`:
+                If return_dict is `True`, [`~schedulers.scheduling_utils.SchedulerOutput`] is returned, otherwise a
+                tuple is returned where 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 self.step_index is None:
+            self._init_step_index(timestep)
+
+        use_corrector = self.step_index > 0 and self.last_sample is not None
+
+        model_output_convert = self.convert_model_output(model_output, sample=sample)
+
+        if use_corrector:
+            current_tau = self.tau_func(self.timestep_list[-1])
+            sample = self.stochastic_adams_moulton_update(
+                this_model_output=model_output_convert,
+                last_sample=self.last_sample,
+                last_noise=self.last_noise,
+                this_sample=sample,
+                order=self.this_corrector_order,
+                tau=current_tau,
+            )
+
+        for i in range(max(self.config.predictor_order, self.config.corrector_order - 1) - 1):
+            self.model_outputs[i] = self.model_outputs[i + 1]
+            self.timestep_list[i] = self.timestep_list[i + 1]
+
+        self.model_outputs[-1] = model_output_convert
+        self.timestep_list[-1] = timestep
+
+        noise = randn_tensor(
+            model_output.shape,
+            generator=generator,
+            device=model_output.device,
+            dtype=model_output.dtype,
+        )
+
+        if self.config.lower_order_final:
+            this_predictor_order = min(self.config.predictor_order, len(self.timesteps) - self.step_index)
+            this_corrector_order = min(self.config.corrector_order, len(self.timesteps) - self.step_index + 1)
+        else:
+            this_predictor_order = self.config.predictor_order
+            this_corrector_order = self.config.corrector_order
+
+        self.this_predictor_order = min(this_predictor_order, self.lower_order_nums + 1)  # warmup for multistep
+        self.this_corrector_order = min(this_corrector_order, self.lower_order_nums + 2)  # warmup for multistep
+        assert self.this_predictor_order > 0
+        assert self.this_corrector_order > 0
+
+        self.last_sample = sample
+        self.last_noise = noise
+
+        current_tau = self.tau_func(self.timestep_list[-1])
+        prev_sample = self.stochastic_adams_bashforth_update(
+            model_output=model_output_convert,
+            sample=sample,
+            noise=noise,
+            order=self.this_predictor_order,
+            tau=current_tau,
+        )
+
+        if self.lower_order_nums < max(self.config.predictor_order, self.config.corrector_order - 1):
+            self.lower_order_nums += 1
+
+        # upon completion increase step index by one
+        self._step_index += 1
+
+        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`):
+                The input sample.
+
+        Returns:
+            `torch.FloatTensor`:
+                A 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

src/diffusers/utils/dummy_pt_objects.py

@@ -990,6 +990,21 @@ class RePaintScheduler(metaclass=DummyObject):
         requires_backends(cls, ["torch"])
 
 
+class SASolverScheduler(metaclass=DummyObject):
+    _backends = ["torch"]
+
+    def __init__(self, *args, **kwargs):
+        requires_backends(self, ["torch"])
+
+    @classmethod
+    def from_config(cls, *args, **kwargs):
+        requires_backends(cls, ["torch"])
+
+    @classmethod
+    def from_pretrained(cls, *args, **kwargs):
+        requires_backends(cls, ["torch"])
+
+
 class SchedulerMixin(metaclass=DummyObject):
     _backends = ["torch"]
 

tests/schedulers/test_scheduler_sasolver.py

+import torch
+
+from diffusers import SASolverScheduler
+from diffusers.utils.testing_utils import require_torchsde, torch_device
+
+from .test_schedulers import SchedulerCommonTest
+
+
+@require_torchsde
+class SASolverSchedulerTest(SchedulerCommonTest):
+    scheduler_classes = (SASolverScheduler,)
+    forward_default_kwargs = (("num_inference_steps", 10),)
+    num_inference_steps = 10
+
+    def get_scheduler_config(self, **kwargs):
+        config = {
+            "num_train_timesteps": 1100,
+            "beta_start": 0.0001,
+            "beta_end": 0.02,
+            "beta_schedule": "linear",
+        }
+
+        config.update(**kwargs)
+        return config
+
+    def test_step_shape(self):
+        kwargs = dict(self.forward_default_kwargs)
+
+        num_inference_steps = kwargs.pop("num_inference_steps", None)
+
+        for scheduler_class in self.scheduler_classes:
+            scheduler_config = self.get_scheduler_config()
+            scheduler = scheduler_class(**scheduler_config)
+
+            sample = self.dummy_sample
+            residual = 0.1 * sample
+
+            if num_inference_steps is not None and hasattr(scheduler, "set_timesteps"):
+                scheduler.set_timesteps(num_inference_steps)
+            elif num_inference_steps is not None and not hasattr(scheduler, "set_timesteps"):
+                kwargs["num_inference_steps"] = num_inference_steps
+
+            # copy over dummy past residuals (must be done after set_timesteps)
+            dummy_past_residuals = [residual + 0.2, residual + 0.15, residual + 0.10]
+            scheduler.model_outputs = dummy_past_residuals[
+                : max(
+                    scheduler.config.predictor_order,
+                    scheduler.config.corrector_order - 1,
+                )
+            ]
+
+            time_step_0 = scheduler.timesteps[5]
+            time_step_1 = scheduler.timesteps[6]
+
+            output_0 = scheduler.step(residual, time_step_0, sample, **kwargs).prev_sample
+            output_1 = scheduler.step(residual, time_step_1, sample, **kwargs).prev_sample
+
+            self.assertEqual(output_0.shape, sample.shape)
+            self.assertEqual(output_0.shape, output_1.shape)
+
+    def test_timesteps(self):
+        for timesteps in [10, 50, 100, 1000]:
+            self.check_over_configs(num_train_timesteps=timesteps)
+
+    def test_betas(self):
+        for beta_start, beta_end in zip([0.00001, 0.0001, 0.001], [0.0002, 0.002, 0.02]):
+            self.check_over_configs(beta_start=beta_start, beta_end=beta_end)
+
+    def test_schedules(self):
+        for schedule in ["linear", "scaled_linear"]:
+            self.check_over_configs(beta_schedule=schedule)
+
+    def test_prediction_type(self):
+        for prediction_type in ["epsilon", "v_prediction"]:
+            self.check_over_configs(prediction_type=prediction_type)
+
+    def test_full_loop_no_noise(self):
+        scheduler_class = self.scheduler_classes[0]
+        scheduler_config = self.get_scheduler_config()
+        scheduler = scheduler_class(**scheduler_config)
+
+        scheduler.set_timesteps(self.num_inference_steps)
+
+        model = self.dummy_model()
+        sample = self.dummy_sample_deter * scheduler.init_noise_sigma
+        sample = sample.to(torch_device)
+        generator = torch.manual_seed(0)
+
+        for i, t in enumerate(scheduler.timesteps):
+            sample = scheduler.scale_model_input(sample, t, generator=generator)
+
+            model_output = model(sample, t)
+
+            output = scheduler.step(model_output, t, sample)
+            sample = output.prev_sample
+
+        result_sum = torch.sum(torch.abs(sample))
+        result_mean = torch.mean(torch.abs(sample))
+
+        if torch_device in ["cpu"]:
+            assert abs(result_sum.item() - 337.394287109375) < 1e-2
+            assert abs(result_mean.item() - 0.43931546807289124) < 1e-3
+        elif torch_device in ["cuda"]:
+            assert abs(result_sum.item() - 329.1999816894531) < 1e-2
+            assert abs(result_mean.item() - 0.4286458194255829) < 1e-3
+        else:
+            print("None")
+
+    def test_full_loop_with_v_prediction(self):
+        scheduler_class = self.scheduler_classes[0]
+        scheduler_config = self.get_scheduler_config(prediction_type="v_prediction")
+        scheduler = scheduler_class(**scheduler_config)
+
+        scheduler.set_timesteps(self.num_inference_steps)
+
+        model = self.dummy_model()
+        sample = self.dummy_sample_deter * scheduler.init_noise_sigma
+        sample = sample.to(torch_device)
+        generator = torch.manual_seed(0)
+
+        for i, t in enumerate(scheduler.timesteps):
+            sample = scheduler.scale_model_input(sample, t, generator=generator)
+
+            model_output = model(sample, t)
+
+            output = scheduler.step(model_output, t, sample)
+            sample = output.prev_sample
+
+        result_sum = torch.sum(torch.abs(sample))
+        result_mean = torch.mean(torch.abs(sample))
+
+        if torch_device in ["cpu"]:
+            assert abs(result_sum.item() - 193.1467742919922) < 1e-2
+            assert abs(result_mean.item() - 0.2514931857585907) < 1e-3
+        elif torch_device in ["cuda"]:
+            assert abs(result_sum.item() - 193.4154052734375) < 1e-2
+            assert abs(result_mean.item() - 0.2518429756164551) < 1e-3
+        else:
+            print("None")
+
+    def test_full_loop_device(self):
+        scheduler_class = self.scheduler_classes[0]
+        scheduler_config = self.get_scheduler_config()
+        scheduler = scheduler_class(**scheduler_config)
+
+        scheduler.set_timesteps(self.num_inference_steps, device=torch_device)
+
+        model = self.dummy_model()
+        sample = self.dummy_sample_deter.to(torch_device) * scheduler.init_noise_sigma
+        generator = torch.manual_seed(0)
+
+        for t in scheduler.timesteps:
+            sample = scheduler.scale_model_input(sample, t)
+
+            model_output = model(sample, t)
+
+            output = scheduler.step(model_output, t, sample, generator=generator)
+            sample = output.prev_sample
+
+        result_sum = torch.sum(torch.abs(sample))
+        result_mean = torch.mean(torch.abs(sample))
+
+        if torch_device in ["cpu"]:
+            assert abs(result_sum.item() - 337.394287109375) < 1e-2
+            assert abs(result_mean.item() - 0.43931546807289124) < 1e-3
+        elif torch_device in ["cuda"]:
+            assert abs(result_sum.item() - 337.394287109375) < 1e-2
+            assert abs(result_mean.item() - 0.4393154978752136) < 1e-3
+        else:
+            print("None")
+
+    def test_full_loop_device_karras_sigmas(self):
+        scheduler_class = self.scheduler_classes[0]
+        scheduler_config = self.get_scheduler_config()
+        scheduler = scheduler_class(**scheduler_config, use_karras_sigmas=True)
+
+        scheduler.set_timesteps(self.num_inference_steps, device=torch_device)
+
+        model = self.dummy_model()
+        sample = self.dummy_sample_deter.to(torch_device) * scheduler.init_noise_sigma
+        sample = sample.to(torch_device)
+        generator = torch.manual_seed(0)
+
+        for t in scheduler.timesteps:
+            sample = scheduler.scale_model_input(sample, t)
+
+            model_output = model(sample, t)
+
+            output = scheduler.step(model_output, t, sample, generator=generator)
+            sample = output.prev_sample
+
+        result_sum = torch.sum(torch.abs(sample))
+        result_mean = torch.mean(torch.abs(sample))
+
+        if torch_device in ["cpu"]:
+            assert abs(result_sum.item() - 837.2554931640625) < 1e-2
+            assert abs(result_mean.item() - 1.0901764631271362) < 1e-2
+        elif torch_device in ["cuda"]:
+            assert abs(result_sum.item() - 837.25537109375) < 1e-2
+            assert abs(result_mean.item() - 1.0901763439178467) < 1e-2
+        else:
+            print("None")