scheduling_pndm.py

# Copyright 2025 Zhejiang University 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: This file is strongly influenced by https://github.com/ermongroup/ddim

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

import numpy as np
import torch

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


# Copied from diffusers.schedulers.scheduling_ddpm.betas_for_alpha_bar
def betas_for_alpha_bar(
    num_diffusion_timesteps: 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 PNDMScheduler(SchedulerMixin, ConfigMixin):
    """
    `PNDMScheduler` uses pseudo numerical methods for diffusion models such as the Runge-Kutta and linear multi-step
    method.

    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`.
        skip_prk_steps (`bool`, defaults to `False`):
            Allows the scheduler to skip the Runge-Kutta steps defined in the original paper as being required before
            PLMS steps.
        set_alpha_to_one (`bool`, defaults to `False`):
            Each diffusion step uses the alphas product value at that step and at the previous one. For the final step
            there is no previous alpha. When this option is `True` the previous alpha product is fixed to `1`,
            otherwise it uses the alpha value at step 0.
        prediction_type (`str`, defaults to `epsilon`, *optional*):
            Prediction type of the scheduler function; can be `epsilon` (predicts the noise of the diffusion process)
            or `v_prediction` (see section 2.4 of [Imagen Video](https://imagen.research.google/video/paper.pdf)
            paper).
        timestep_spacing (`str`, defaults to `"leading"`):
            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 = 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,
        skip_prk_steps: bool = False,
        set_alpha_to_one: bool = False,
        prediction_type: str = "epsilon",
        timestep_spacing: str = "leading",
        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} is not implemented for {self.__class__}")

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

        self.final_alpha_cumprod = torch.tensor(1.0) if set_alpha_to_one else self.alphas_cumprod[0]

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

        # For now we only support F-PNDM, i.e. the runge-kutta method
        # For more information on the algorithm please take a look at the paper: https://huggingface.co/papers/2202.09778
        # mainly at formula (9), (12), (13) and the Algorithm 2.
        self.pndm_order = 4

        # running values
        self.cur_model_output = 0
        self.counter = 0
        self.cur_sample = None
        self.ets = []

        # setable values
        self.num_inference_steps = None
        self._timesteps = np.arange(0, num_train_timesteps)[::-1].copy()
        self.prk_timesteps = None
        self.plms_timesteps = None
        self.timesteps = None

    def set_timesteps(self, num_inference_steps: int, 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.
        """

        self.num_inference_steps = num_inference_steps
        # "linspace", "leading", "trailing" corresponds to annotation of Table 2. of https://huggingface.co/papers/2305.08891
        if self.config.timestep_spacing == "linspace":
            self._timesteps = (
                np.linspace(0, self.config.num_train_timesteps - 1, num_inference_steps).round().astype(np.int64)
            )
        elif self.config.timestep_spacing == "leading":
            step_ratio = self.config.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
            self._timesteps = (np.arange(0, num_inference_steps) * step_ratio).round()
            self._timesteps += self.config.steps_offset
        elif self.config.timestep_spacing == "trailing":
            step_ratio = self.config.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
            self._timesteps = np.round(np.arange(self.config.num_train_timesteps, 0, -step_ratio))[::-1].astype(
                np.int64
            )
            self._timesteps -= 1
        else:
            raise ValueError(
                f"{self.config.timestep_spacing} is not supported. Please make sure to choose one of 'linspace', 'leading' or 'trailing'."
            )

        if self.config.skip_prk_steps:
            # for some models like stable diffusion the prk steps can/should be skipped to
            # produce better results. When using PNDM with `self.config.skip_prk_steps` the implementation
            # is based on crowsonkb's PLMS sampler implementation: https://github.com/CompVis/latent-diffusion/pull/51
            self.prk_timesteps = np.array([])
            self.plms_timesteps = np.concatenate([self._timesteps[:-1], self._timesteps[-2:-1], self._timesteps[-1:]])[
                ::-1
            ].copy()
        else:
            prk_timesteps = np.array(self._timesteps[-self.pndm_order :]).repeat(2) + np.tile(
                np.array([0, self.config.num_train_timesteps // num_inference_steps // 2]), self.pndm_order
            )
            self.prk_timesteps = (prk_timesteps[:-1].repeat(2)[1:-1])[::-1].copy()
            self.plms_timesteps = self._timesteps[:-3][
                ::-1
            ].copy()  # we copy to avoid having negative strides which are not supported by torch.from_numpy

        timesteps = np.concatenate([self.prk_timesteps, self.plms_timesteps]).astype(np.int64)
        self.timesteps = torch.from_numpy(timesteps).to(device)

        self.ets = []
        self.counter = 0
        self.cur_model_output = 0

    def step(
        self,
        model_output: torch.Tensor,
        timestep: int,
        sample: torch.Tensor,
        return_dict: bool = True,
    ) -> Union[SchedulerOutput, 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), and calls [`~PNDMScheduler.step_prk`]
        or [`~PNDMScheduler.step_plms`] depending on the internal variable `counter`.

        Args:
            model_output (`torch.Tensor`):
                The direct output from learned diffusion model.
            timestep (`int`):
                The current discrete timestep in the diffusion chain.
            sample (`torch.Tensor`):
                A current instance of a sample created by the diffusion process.
            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.counter < len(self.prk_timesteps) and not self.config.skip_prk_steps:
            return self.step_prk(model_output=model_output, timestep=timestep, sample=sample, return_dict=return_dict)
        else:
            return self.step_plms(model_output=model_output, timestep=timestep, sample=sample, return_dict=return_dict)

    def step_prk(
        self,
        model_output: torch.Tensor,
        timestep: int,
        sample: torch.Tensor,
        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 Runge-Kutta method. It performs four forward passes to approximate the solution to the differential
        equation.

        Args:
            model_output (`torch.Tensor`):
                The direct output from learned diffusion model.
            timestep (`int`):
                The current discrete timestep in the diffusion chain.
            sample (`torch.Tensor`):
                A current instance of a sample created by the diffusion process.
            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"
            )

        diff_to_prev = 0 if self.counter % 2 else self.config.num_train_timesteps // self.num_inference_steps // 2
        prev_timestep = timestep - diff_to_prev
        timestep = self.prk_timesteps[self.counter // 4 * 4]

        if self.counter % 4 == 0:
            self.cur_model_output += 1 / 6 * model_output
            self.ets.append(model_output)
            self.cur_sample = sample
        elif (self.counter - 1) % 4 == 0:
            self.cur_model_output += 1 / 3 * model_output
        elif (self.counter - 2) % 4 == 0:
            self.cur_model_output += 1 / 3 * model_output
        elif (self.counter - 3) % 4 == 0:
            model_output = self.cur_model_output + 1 / 6 * model_output
            self.cur_model_output = 0

        # cur_sample should not be `None`
        cur_sample = self.cur_sample if self.cur_sample is not None else sample

        prev_sample = self._get_prev_sample(cur_sample, timestep, prev_timestep, model_output)
        self.counter += 1

        if not return_dict:
            return (prev_sample,)

        return SchedulerOutput(prev_sample=prev_sample)

    def step_plms(
        self,
        model_output: torch.Tensor,
        timestep: int,
        sample: torch.Tensor,
        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 linear multistep method. It performs one forward pass multiple times to approximate the solution.

        Args:
            model_output (`torch.Tensor`):
                The direct output from learned diffusion model.
            timestep (`int`):
                The current discrete timestep in the diffusion chain.
            sample (`torch.Tensor`):
                A current instance of a sample created by the diffusion process.
            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 not self.config.skip_prk_steps and len(self.ets) < 3:
            raise ValueError(
                f"{self.__class__} can only be run AFTER scheduler has been run "
                "in 'prk' mode for at least 12 iterations "
                "See: https://github.com/huggingface/diffusers/blob/main/src/diffusers/pipelines/pipeline_pndm.py "
                "for more information."
            )

        prev_timestep = timestep - self.config.num_train_timesteps // self.num_inference_steps

        if self.counter != 1:
            self.ets = self.ets[-3:]
            self.ets.append(model_output)
        else:
            prev_timestep = timestep
            timestep = timestep + self.config.num_train_timesteps // self.num_inference_steps

        if len(self.ets) == 1 and self.counter == 0:
            model_output = model_output
            self.cur_sample = sample
        elif len(self.ets) == 1 and self.counter == 1:
            model_output = (model_output + self.ets[-1]) / 2
            sample = self.cur_sample
            self.cur_sample = None
        elif len(self.ets) == 2:
            model_output = (3 * self.ets[-1] - self.ets[-2]) / 2
        elif len(self.ets) == 3:
            model_output = (23 * self.ets[-1] - 16 * self.ets[-2] + 5 * self.ets[-3]) / 12
        else:
            model_output = (1 / 24) * (55 * self.ets[-1] - 59 * self.ets[-2] + 37 * self.ets[-3] - 9 * self.ets[-4])

        prev_sample = self._get_prev_sample(sample, timestep, prev_timestep, model_output)
        self.counter += 1

        if not return_dict:
            return (prev_sample,)

        return SchedulerOutput(prev_sample=prev_sample)

    def scale_model_input(self, sample: torch.Tensor, *args, **kwargs) -> 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.

        Returns:
            `torch.Tensor`:
                A scaled input sample.
        """
        return sample

    def _get_prev_sample(self, sample, timestep, prev_timestep, model_output):
        # See formula (9) of PNDM paper https://huggingface.co/papers/2202.09778
        # this function computes x_(t−δ) using the formula of (9)
        # Note that x_t needs to be added to both sides of the equation

        # Notation (<variable name> -> <name in paper>
        # alpha_prod_t -> α_t
        # alpha_prod_t_prev -> α_(t−δ)
        # beta_prod_t -> (1 - α_t)
        # beta_prod_t_prev -> (1 - α_(t−δ))
        # sample -> x_t
        # model_output -> e_θ(x_t, t)
        # prev_sample -> x_(t−δ)
        alpha_prod_t = self.alphas_cumprod[timestep]
        alpha_prod_t_prev = self.alphas_cumprod[prev_timestep] if prev_timestep >= 0 else self.final_alpha_cumprod
        beta_prod_t = 1 - alpha_prod_t
        beta_prod_t_prev = 1 - alpha_prod_t_prev

        if self.config.prediction_type == "v_prediction":
            model_output = (alpha_prod_t**0.5) * model_output + (beta_prod_t**0.5) * sample
        elif self.config.prediction_type != "epsilon":
            raise ValueError(
                f"prediction_type given as {self.config.prediction_type} must be one of `epsilon` or `v_prediction`"
            )

        # corresponds to (α_(t−δ) - α_t) divided by
        # denominator of x_t in formula (9) and plus 1
        # Note: (α_(t−δ) - α_t) / (sqrt(α_t) * (sqrt(α_(t−δ)) + sqr(α_t))) =
        # sqrt(α_(t−δ)) / sqrt(α_t))
        sample_coeff = (alpha_prod_t_prev / alpha_prod_t) ** (0.5)

        # corresponds to denominator of e_θ(x_t, t) in formula (9)
        model_output_denom_coeff = alpha_prod_t * beta_prod_t_prev ** (0.5) + (
            alpha_prod_t * beta_prod_t * alpha_prod_t_prev
        ) ** (0.5)

        # full formula (9)
        prev_sample = (
            sample_coeff * sample - (alpha_prod_t_prev - alpha_prod_t) * model_output / model_output_denom_coeff
        )

        return prev_sample

    # Copied from diffusers.schedulers.scheduling_ddpm.DDPMScheduler.add_noise
    def add_noise(
        self,
        original_samples: torch.Tensor,
        noise: torch.Tensor,
        timesteps: torch.IntTensor,
    ) -> torch.Tensor:
        """
        Add noise to the original samples according to the noise magnitude at each timestep (this is the forward
        diffusion process).

        Args:
            original_samples (`torch.Tensor`):
                The original samples to which noise will be added.
            noise (`torch.Tensor`):
                The noise to add to the samples.
            timesteps (`torch.IntTensor`):
                The timesteps indicating the noise level for each sample.

        Returns:
            `torch.Tensor`:
                The noisy samples.
        """
        # Make sure alphas_cumprod and timestep have same device and dtype as original_samples
        # Move the self.alphas_cumprod to device to avoid redundant CPU to GPU data movement
        # for the subsequent add_noise calls
        self.alphas_cumprod = self.alphas_cumprod.to(device=original_samples.device)
        alphas_cumprod = self.alphas_cumprod.to(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