webUI_ExtraSchedulers/scripts/res_solver.py

import torch
from torch import no_grad, FloatTensor
from tqdm import tqdm
from itertools import pairwise
from typing import Protocol, Optional, Dict, Any, TypedDict, NamedTuple, Union, List
import math

from tqdm.auto import trange

#   copied from kdiffusion/sampling.py and utils.py
def default_noise_sampler(x):
    return lambda sigma, sigma_next: torch.randn_like(x)
def append_dims(x, target_dims):
    """Appends dimensions to the end of a tensor until it has target_dims dimensions."""
    dims_to_append = target_dims - x.ndim
    if dims_to_append < 0:
        raise ValueError(f'input has {x.ndim} dims but target_dims is {target_dims}, which is less')
    return x[(...,) + (None,) * dims_to_append]
def to_d(x, sigma, denoised):
    """Converts a denoiser output to a Karras ODE derivative."""
    return (x - denoised) / append_dims(sigma, x.ndim)


class DenoiserModel(Protocol):
  def __call__(self, x: FloatTensor, t: FloatTensor, *args, **kwargs) -> FloatTensor: ...

class RefinedExpCallbackPayload(TypedDict):
  x: FloatTensor
  i: int
  sigma: FloatTensor
  sigma_hat: FloatTensor

class RefinedExpCallback(Protocol):
  def __call__(self, payload: RefinedExpCallbackPayload) -> None: ...

class NoiseSampler(Protocol):
  def __call__(self, x: FloatTensor) -> FloatTensor: ...

class StepOutput(NamedTuple):
  x_next: FloatTensor
  denoised: FloatTensor
  denoised2: FloatTensor
  vel: FloatTensor
  vel_2: FloatTensor

def _gamma(

  n: int,

) -> int:
  """

  https://en.wikipedia.org/wiki/Gamma_function

  for every positive integer n,

  Γ(n) = (n-1)!

  """
  return math.factorial(n-1)

def _incomplete_gamma(

  s: int,

  x: float,

  gamma_s: Optional[int] = None

) -> float:
  """

  https://en.wikipedia.org/wiki/Incomplete_gamma_function#Special_values

  if s is a positive integer,

  Γ(s, x) = (s-1)!*∑{k=0..s-1}(x^k/k!)

  """
  if gamma_s is None:
    gamma_s = _gamma(s)

  sum_: float = 0
  # {k=0..s-1} inclusive
  for k in range(s):
    numerator: float = x**k
    denom: int = math.factorial(k)
    quotient: float = numerator/denom
    sum_ += quotient
  incomplete_gamma_: float = sum_ * math.exp(-x) * gamma_s
  return incomplete_gamma_

# by Katherine Crowson
def _phi_1(neg_h: FloatTensor):
  return torch.nan_to_num(torch.expm1(neg_h) / neg_h, nan=1.0)

# by Katherine Crowson
def _phi_2(neg_h: FloatTensor):
  return torch.nan_to_num((torch.expm1(neg_h) - neg_h) / neg_h**2, nan=0.5)

# by Katherine Crowson
def _phi_3(neg_h: FloatTensor):
  return torch.nan_to_num((torch.expm1(neg_h) - neg_h - neg_h**2 / 2) / neg_h**3, nan=1 / 6)

def _phi(

  neg_h: float,

  j: int,

):
  """

  For j={1,2,3}: you could alternatively use Kat's phi_1, phi_2, phi_3 which perform fewer steps



  Lemma 1

  https://arxiv.org/abs/2308.02157

  ϕj(-h) = 1/h^j*∫{0..h}(e^(τ-h)*(τ^(j-1))/((j-1)!)dτ)



  https://www.wolframalpha.com/input?i=integrate+e%5E%28%CF%84-h%29*%28%CF%84%5E%28j-1%29%2F%28j-1%29%21%29d%CF%84

  = 1/h^j*[(e^(-h)*(-τ)^(-j)*τ(j))/((j-1)!)]{0..h}

  https://www.wolframalpha.com/input?i=integrate+e%5E%28%CF%84-h%29*%28%CF%84%5E%28j-1%29%2F%28j-1%29%21%29d%CF%84+between+0+and+h

  = 1/h^j*((e^(-h)*(-h)^(-j)*h^j*(Γ(j)-Γ(j,-h)))/(j-1)!)

  = (e^(-h)*(-h)^(-j)*h^j*(Γ(j)-Γ(j,-h))/((j-1)!*h^j)

  = (e^(-h)*(-h)^(-j)*(Γ(j)-Γ(j,-h))/(j-1)!

  = (e^(-h)*(-h)^(-j)*(Γ(j)-Γ(j,-h))/Γ(j)

  = (e^(-h)*(-h)^(-j)*(1-Γ(j,-h)/Γ(j))



  requires j>0

  """
  assert j > 0
  gamma_: float = _gamma(j)
  incomp_gamma_: float = _incomplete_gamma(j, neg_h, gamma_s=gamma_)

  phi_: float = math.exp(neg_h) * neg_h**-j * (1-incomp_gamma_/gamma_)

  return phi_

class RESDECoeffsSecondOrder(NamedTuple):
  a2_1: float
  b1: float
  b2: float

def _de_second_order(

  h: float,

  c2: float,

  simple_phi_calc = False,

) -> RESDECoeffsSecondOrder:
  """

  Table 3

  https://arxiv.org/abs/2308.02157

  ϕi,j := ϕi,j(-h) = ϕi(-cj*h)

  a2_1 = c2ϕ1,2

       = c2ϕ1(-c2*h)

  b1 = ϕ1 - ϕ2/c2

  """
  if simple_phi_calc:
    # Kat computed simpler expressions for phi for cases j={1,2,3}
    a2_1: float = c2 * _phi_1(-c2*h)
    phi1: float = _phi_1(-h)
    phi2: float = _phi_2(-h)
  else:
    # I computed general solution instead.
    # they're close, but there are slight differences. not sure which would be more prone to numerical error.
    a2_1: float = c2 * _phi(j=1, neg_h=-c2*h)
    phi1: float = _phi(j=1, neg_h=-h)
    phi2: float = _phi(j=2, neg_h=-h)
  phi2_c2: float = phi2/c2
  b1: float = phi1 - phi2_c2
  b2: float = phi2_c2
  return RESDECoeffsSecondOrder(
    a2_1=a2_1,
    b1=b1,
    b2=b2,
  )  

def _refined_exp_sosu_step(

  model: DenoiserModel,

  x: FloatTensor,

  sigma: FloatTensor,

  sigma_next: FloatTensor,

  c2 = 0.5,

  extra_args: Dict[str, Any] = {},

  pbar: Optional[tqdm] = None,

  simple_phi_calc = False,

  momentum = 0.0,

  vel = None,

  vel_2 = None,

  time = None

) -> StepOutput:
  """

  Algorithm 1 "RES Second order Single Update Step with c2"

  https://arxiv.org/abs/2308.02157



  Parameters:

    model (`DenoiserModel`): a k-diffusion wrapped denoiser model (e.g. a subclass of DiscreteEpsDDPMDenoiser)

    x (`FloatTensor`): noised latents (or RGB I suppose), e.g. torch.randn((B, C, H, W)) * sigma[0]

    sigma (`FloatTensor`): timestep to denoise

    sigma_next (`FloatTensor`): timestep+1 to denoise

    c2 (`float`, *optional*, defaults to .5): partial step size for solving ODE. .5 = midpoint method

    extra_args (`Dict[str, Any]`, *optional*, defaults to `{}`): kwargs to pass to `model#__call__()`

    pbar (`tqdm`, *optional*, defaults to `None`): progress bar to update after each model call

    simple_phi_calc (`bool`, *optional*, defaults to `True`): True = calculate phi_i,j(-h) via simplified formulae specific to j={1,2}. False = Use general solution that works for any j. Mathematically equivalent, but could be numeric differences.

  """

  def momentum_func(diff, velocity, timescale=1.0, offset=-momentum / 2.0): # Diff is current diff, vel is previous diff
    if velocity is None:
        momentum_vel = diff
    else:
        momentum_vel = momentum * (timescale + offset) * velocity + (1 - momentum * (timescale + offset)) * diff
    return momentum_vel

  lam_next, lam = (s.log().neg() for s in (sigma_next, sigma))

  # type hints aren't strictly true regarding float vs FloatTensor.
  # everything gets promoted to `FloatTensor` after interacting with `sigma: FloatTensor`.
  # I will use float to indicate any variables which are scalars.
  h: float = lam_next - lam
  a2_1, b1, b2 = _de_second_order(h=h, c2=c2, simple_phi_calc=simple_phi_calc)
  
  denoised: FloatTensor = model(x, sigma.repeat(x.size(0)), **extra_args)
  # if pbar is not None:
    # pbar.update(0.5)

  c2_h: float = c2*h

  diff_2 = momentum_func(a2_1*h*denoised, vel_2, time)
  vel_2 = diff_2
  x_2: FloatTensor = math.exp(-c2_h)*x + diff_2
  lam_2: float = lam + c2_h
  sigma_2: float = lam_2.neg().exp()

  denoised2: FloatTensor = model(x_2, sigma_2.repeat(x_2.size(0)), **extra_args)
  if pbar is not None:
    pbar.update()

  diff = momentum_func(h*(b1*denoised + b2*denoised2), vel, time)
  vel = diff

  x_next: FloatTensor = math.exp(-h)*x + diff
  
  return StepOutput(
    x_next=x_next,
    denoised=denoised,
    denoised2=denoised2,
    vel=vel,
    vel_2=vel_2,
  )
  

@no_grad()
def sample_refined_exp_s(

  model: FloatTensor,

  x: FloatTensor,

  sigmas: FloatTensor,

  denoise_to_zero: bool = True,

  extra_args: Dict[str, Any] = {},

  callback: Optional[RefinedExpCallback] = None,

  disable: Optional[bool] = None,

  ita: FloatTensor = torch.zeros((1,)),

  c2 = .5,

  noise_sampler: NoiseSampler = torch.randn_like,

  simple_phi_calc = False,

  momentum = 0.0,

):
  """

  Refined Exponential Solver (S).

  Algorithm 2 "RES Single-Step Sampler" with Algorithm 1 second-order step

  https://arxiv.org/abs/2308.02157



  Parameters:

    model (`DenoiserModel`): a k-diffusion wrapped denoiser model (e.g. a subclass of DiscreteEpsDDPMDenoiser)

    x (`FloatTensor`): noised latents (or RGB I suppose), e.g. torch.randn((B, C, H, W)) * sigma[0]

    sigmas (`FloatTensor`): sigmas (ideally an exponential schedule!) e.g. get_sigmas_exponential(n=25, sigma_min=model.sigma_min, sigma_max=model.sigma_max)

    denoise_to_zero (`bool`, *optional*, defaults to `True`): whether to finish with a first-order step down to 0 (rather than stopping at sigma_min). True = fully denoise image. False = match Algorithm 2 in paper

    extra_args (`Dict[str, Any]`, *optional*, defaults to `{}`): kwargs to pass to `model#__call__()`

    callback (`RefinedExpCallback`, *optional*, defaults to `None`): you can supply this callback to see the intermediate denoising results, e.g. to preview each step of the denoising process

    disable (`bool`, *optional*, defaults to `False`): whether to hide `tqdm`'s progress bar animation from being printed

    ita (`FloatTensor`, *optional*, defaults to 0.): degree of stochasticity, η, for each timestep. tensor shape must be broadcastable to 1-dimensional tensor with length `len(sigmas) if denoise_to_zero else len(sigmas)-1`. each element should be from 0 to 1.

         - if used: batch noise doesn't match non-batch

    c2 (`float`, *optional*, defaults to .5): partial step size for solving ODE. .5 = midpoint method

    noise_sampler (`NoiseSampler`, *optional*, defaults to `torch.randn_like`): method used for adding noise

    simple_phi_calc (`bool`, *optional*, defaults to `True`): True = calculate phi_i,j(-h) via simplified formulae specific to j={1,2}. False = Use general solution that works for any j. Mathematically equivalent, but could be numeric differences.

  """
  #assert sigmas[-1] == 0
  device = x.device
  ita = ita.to(device)
  sigmas = sigmas.to(device)

  sigma_min, sigma_max = sigmas[sigmas > 0].min(), sigmas.max()

  vel, vel_2 = None, None
  with tqdm(disable=disable, total=len(sigmas)-(1 if denoise_to_zero else 2)) as pbar:
    for i, (sigma, sigma_next) in enumerate(pairwise(sigmas[:-1].split(1))):
      time = sigmas[i] / sigma_max
      if 'sigma' not in locals():
        sigma = sigmas[i]
      eps = torch.randn_like(x).float()
      sigma_hat = sigma * (1 + ita)
      x_hat = x + (sigma_hat ** 2 - sigma ** 2).sqrt() * eps
      x_next, denoised, denoised2, vel, vel_2 = _refined_exp_sosu_step(
        model,
        x_hat,
        sigma_hat,
        sigma_next,
        c2=c2,
        extra_args=extra_args,
        pbar=pbar,
        simple_phi_calc=simple_phi_calc,
        momentum = momentum,
        vel = vel,
        vel_2 = vel_2,
        time = time
      )
      if callback is not None:
        payload = RefinedExpCallbackPayload(
          x=x,
          i=i,
          sigma=sigma,
          sigma_hat=sigma_hat,
          denoised=denoised,
          denoised2=denoised2,
        )
        callback(payload)
      x = x_next
    if denoise_to_zero:
      eps = torch.randn_like(x).float()
      sigma_hat = sigma * (1 + ita)
      x_hat = x + (sigma_hat ** 2 - sigma ** 2).sqrt() * eps
      x_next: FloatTensor = model(x_hat, sigma.to(x_hat.device).repeat(x_hat.size(0)), **extra_args)
      pbar.update()

      if callback is not None:
        payload = RefinedExpCallbackPayload(
          x=x,
          i=i,
          sigma=sigma,
          sigma_hat=sigma_hat,
          denoised=denoised,
          denoised2=denoised2,
        )
        callback(payload)


      x = x_next
  return x

# Many thanks to Kat + Birch-San for this wonderful sampler implementation! https://github.com/Birch-san/sdxl-play/commits/res/
def sample_res_solver(model, x, sigmas, extra_args=None, callback=None, disable=None, noise_sampler_type="gaussian", noise_sampler=None, denoise_to_zero=True, simple_phi_calc=False, c2=0.5, ita=torch.Tensor((0.0,)), momentum=0.0):
    return sample_refined_exp_s(model, x, sigmas, extra_args=extra_args, callback=callback, disable=disable, noise_sampler=noise_sampler, denoise_to_zero=denoise_to_zero, simple_phi_calc=simple_phi_calc, c2=c2, ita=ita, momentum=momentum)


##  modified from ReForge, original implementation ComfyUI
@torch.no_grad()
def res_multistep(model, x, sigmas, extra_args=None, callback=None, disable=None, s_churn=0., s_tmin=0., s_tmax=float('inf'), s_noise=1., noise_sampler=None, cfgpp=False):
    extra_args = {} if extra_args is None else extra_args
    seed = extra_args.get("seed", None)
    noise_sampler = default_noise_sampler(x) if noise_sampler is None else noise_sampler
    s_in = x.new_ones([x.shape[0]])
    sigma_fn = lambda t: t.neg().exp()
    t_fn = lambda sigma: sigma.log().neg()
    phi1_fn = lambda t: torch.expm1(t) / t
    phi2_fn = lambda t: (phi1_fn(t) - 1.0) / t
    old_denoised = None

    sigmas = sigmas.to(x.device)

    if cfgpp:
        model.need_last_noise_uncond = True
        model.inner_model.inner_model.forge_objects.unet.model_options["disable_cfg1_optimization"] = True

    for i in trange(len(sigmas) - 1, disable=disable):
        if s_churn > 0:
            gamma = min(s_churn / (len(sigmas) - 1), 2**0.5 - 1) if s_tmin <= sigmas[i] <= s_tmax else 0.0
            sigma_hat = sigmas[i] * (gamma + 1)
        else:
            gamma = 0
            sigma_hat = sigmas[i]
        if gamma > 0:
            eps = torch.randn_like(x) * s_noise
            x = x + eps * (sigma_hat**2 - sigmas[i] ** 2) ** 0.5
        denoised = model(x, sigma_hat * s_in, **extra_args)

        if callback is not None:
            callback({"x": x, "i": i, "sigma": sigmas[i], "sigma_hat": sigma_hat, "denoised": denoised})
        if sigmas[i + 1] == 0 or old_denoised is None:
            # Euler method
            if cfgpp:
                d = model.last_noise_uncond
                x = denoised + d * sigmas[i + 1]
            else:
                d = to_d(x, sigma_hat, denoised)
                dt = sigmas[i + 1] - sigma_hat
                x = x + d * dt
        else:
            # Second order multistep method in https://arxiv.org/pdf/2308.02157
            t, t_next, t_prev = t_fn(sigmas[i]), t_fn(sigmas[i + 1]), t_fn(sigmas[i - 1])
            h = t_next - t
            c2 = (t_prev - t) / h
            phi1_val, phi2_val = phi1_fn(-h), phi2_fn(-h)
            b1 = torch.nan_to_num(phi1_val - 1.0 / c2 * phi2_val, nan=0.0)
            b2 = torch.nan_to_num(1.0 / c2 * phi2_val, nan=0.0)
            if cfgpp:
                d = model.last_noise_uncond
                x = denoised + d * sigma_hat

            x = (sigma_fn(t_next) / sigma_fn(t)) * x + h * (b1 * denoised + b2 * old_denoised)
        old_denoised = denoised
    return x
@torch.no_grad()
def sample_res_multistep(model, x, sigmas, extra_args=None, callback=None, disable=None, s_churn=0., s_tmin=0., s_tmax=float('inf'), s_noise=1., noise_sampler=None):
    return res_multistep(model, x, sigmas, extra_args=extra_args, callback=callback, disable=disable, s_churn=s_churn, s_tmin=s_tmin, s_tmax=s_tmax, s_noise=s_noise, noise_sampler=noise_sampler, cfgpp=False)
@torch.no_grad()
def sample_res_multistep_cfgpp(model, x, sigmas, extra_args=None, callback=None, disable=None, s_churn=0., s_tmin=0., s_tmax=float('inf'), s_noise=1., noise_sampler=None):
    return res_multistep(model, x, sigmas, extra_args=extra_args, callback=callback, disable=disable, s_churn=s_churn, s_tmin=s_tmin, s_tmax=s_tmax, s_noise=s_noise, noise_sampler=noise_sampler, cfgpp=True)