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sampling.py

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+import math
+
+from scipy import integrate
+import torch
+from torch import nn
+from torchdiffeq import odeint
+import torchsde
+from tqdm.auto import trange, tqdm
+
+from . import utils
+
+
+def append_zero(x):
+    return torch.cat([x, x.new_zeros([1])])
+
+
+def get_sigmas_karras(n, sigma_min, sigma_max, rho=7., device='cpu'):
+    """Constructs the noise schedule of Karras et al. (2022)."""
+    ramp = torch.linspace(0, 1, n)
+    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 append_zero(sigmas).to(device)
+
+
+def get_sigmas_exponential(n, sigma_min, sigma_max, device='cpu'):
+    """Constructs an exponential noise schedule."""
+    sigmas = torch.linspace(math.log(sigma_max), math.log(sigma_min), n, device=device).exp()
+    return append_zero(sigmas)
+
+
+def get_sigmas_polyexponential(n, sigma_min, sigma_max, rho=1., device='cpu'):
+    """Constructs an polynomial in log sigma noise schedule."""
+    ramp = torch.linspace(1, 0, n, device=device) ** rho
+    sigmas = torch.exp(ramp * (math.log(sigma_max) - math.log(sigma_min)) + math.log(sigma_min))
+    return append_zero(sigmas)
+
+
+def get_sigmas_vp(n, beta_d=19.9, beta_min=0.1, eps_s=1e-3, device='cpu'):
+    """Constructs a continuous VP noise schedule."""
+    t = torch.linspace(1, eps_s, n, device=device)
+    sigmas = torch.sqrt(torch.exp(beta_d * t ** 2 / 2 + beta_min * t) - 1)
+    return append_zero(sigmas)
+
+
+def to_d(x, sigma, denoised):
+    """Converts a denoiser output to a Karras ODE derivative."""
+    return (x - denoised) / utils.append_dims(sigma, x.ndim)
+
+
+def get_ancestral_step(sigma_from, sigma_to, eta=1.):
+    """Calculates the noise level (sigma_down) to step down to and the amount
+    of noise to add (sigma_up) when doing an ancestral sampling step."""
+    if not eta:
+        return sigma_to, 0.
+    sigma_up = min(sigma_to, eta * (sigma_to ** 2 * (sigma_from ** 2 - sigma_to ** 2) / sigma_from ** 2) ** 0.5)
+    sigma_down = (sigma_to ** 2 - sigma_up ** 2) ** 0.5
+    return sigma_down, sigma_up
+
+
+def default_noise_sampler(x):
+    return lambda sigma, sigma_next: torch.randn_like(x)
+
+
+class BatchedBrownianTree:
+    """A wrapper around torchsde.BrownianTree that enables batches of entropy."""
+
+    def __init__(self, x, t0, t1, seed=None, **kwargs):
+        t0, t1, self.sign = self.sort(t0, t1)
+        w0 = kwargs.get('w0', torch.zeros_like(x))
+        if seed is None:
+            seed = torch.randint(0, 2 ** 63 - 1, []).item()
+        self.batched = True
+        try:
+            assert len(seed) == x.shape[0]
+            w0 = w0[0]
+        except TypeError:
+            seed = [seed]
+            self.batched = False
+        self.trees = [torchsde.BrownianTree(t0, w0, t1, entropy=s, **kwargs) for s in seed]
+
+    @staticmethod
+    def sort(a, b):
+        return (a, b, 1) if a < b else (b, a, -1)
+
+    def __call__(self, t0, t1):
+        t0, t1, sign = self.sort(t0, t1)
+        w = torch.stack([tree(t0, t1) for tree in self.trees]) * (self.sign * sign)
+        return w if self.batched else w[0]
+
+
+class BrownianTreeNoiseSampler:
+    """A noise sampler backed by a torchsde.BrownianTree.
+
+    Args:
+        x (Tensor): The tensor whose shape, device and dtype to use to generate
+            random samples.
+        sigma_min (float): The low end of the valid interval.
+        sigma_max (float): The high end of the valid interval.
+        seed (int or List[int]): The random seed. If a list of seeds is
+            supplied instead of a single integer, then the noise sampler will
+            use one BrownianTree per batch item, each with its own seed.
+        transform (callable): A function that maps sigma to the sampler's
+            internal timestep.
+    """
+
+    def __init__(self, x, sigma_min, sigma_max, seed=None, transform=lambda x: x):
+        self.transform = transform
+        t0, t1 = self.transform(torch.as_tensor(sigma_min)), self.transform(torch.as_tensor(sigma_max))
+        self.tree = BatchedBrownianTree(x, t0, t1, seed)
+
+    def __call__(self, sigma, sigma_next):
+        t0, t1 = self.transform(torch.as_tensor(sigma)), self.transform(torch.as_tensor(sigma_next))
+        return self.tree(t0, t1) / (t1 - t0).abs().sqrt()
+
+
+@torch.no_grad()
+def sample_euler(model, x, sigmas, extra_args=None, callback=None, disable=None, s_churn=0., s_tmin=0., s_tmax=float('inf'), s_noise=1.):
+    """Implements Algorithm 2 (Euler steps) from Karras et al. (2022)."""
+    extra_args = {} if extra_args is None else extra_args
+    s_in = x.new_ones([x.shape[0]])
+    for i in trange(len(sigmas) - 1, disable=disable):
+        gamma = min(s_churn / (len(sigmas) - 1), 2 ** 0.5 - 1) if s_tmin <= sigmas[i] <= s_tmax else 0.
+        eps = torch.randn_like(x) * s_noise
+        sigma_hat = sigmas[i] * (gamma + 1)
+        if gamma > 0:
+            x = x + eps * (sigma_hat ** 2 - sigmas[i] ** 2) ** 0.5
+        denoised = model(x, sigma_hat * s_in, **extra_args)
+        d = to_d(x, sigma_hat, denoised)
+        if callback is not None:
+            callback({'x': x, 'i': i, 'sigma': sigmas[i], 'sigma_hat': sigma_hat, 'denoised': denoised})
+        dt = sigmas[i + 1] - sigma_hat
+        # Euler method
+        x = x + d * dt
+    return x
+
+
+@torch.no_grad()
+def sample_euler_ancestral(model, x, sigmas, extra_args=None, callback=None, disable=None, eta=1., s_noise=1., noise_sampler=None):
+    """Ancestral sampling with Euler method steps."""
+    extra_args = {} if extra_args is None else extra_args
+    noise_sampler = default_noise_sampler(x) if noise_sampler is None else noise_sampler
+    s_in = x.new_ones([x.shape[0]])
+    for i in trange(len(sigmas) - 1, disable=disable):
+        denoised = model(x, sigmas[i] * s_in, **extra_args)
+        sigma_down, sigma_up = get_ancestral_step(sigmas[i], sigmas[i + 1], eta=eta)
+        if callback is not None:
+            callback({'x': x, 'i': i, 'sigma': sigmas[i], 'sigma_hat': sigmas[i], 'denoised': denoised})
+        d = to_d(x, sigmas[i], denoised)
+        # Euler method
+        dt = sigma_down - sigmas[i]
+        x = x + d * dt
+        if sigmas[i + 1] > 0:
+            x = x + noise_sampler(sigmas[i], sigmas[i + 1]) * s_noise * sigma_up
+    return x
+
+
+@torch.no_grad()
+def sample_heun(model, x, sigmas, extra_args=None, callback=None, disable=None, s_churn=0., s_tmin=0., s_tmax=float('inf'), s_noise=1.):
+    """Implements Algorithm 2 (Heun steps) from Karras et al. (2022)."""
+    extra_args = {} if extra_args is None else extra_args
+    s_in = x.new_ones([x.shape[0]])
+    for i in trange(len(sigmas) - 1, disable=disable):
+        gamma = min(s_churn / (len(sigmas) - 1), 2 ** 0.5 - 1) if s_tmin <= sigmas[i] <= s_tmax else 0.
+        eps = torch.randn_like(x) * s_noise
+        sigma_hat = sigmas[i] * (gamma + 1)
+        if gamma > 0:
+            x = x + eps * (sigma_hat ** 2 - sigmas[i] ** 2) ** 0.5
+        denoised = model(x, sigma_hat * s_in, **extra_args)
+        d = to_d(x, sigma_hat, denoised)
+        if callback is not None:
+            callback({'x': x, 'i': i, 'sigma': sigmas[i], 'sigma_hat': sigma_hat, 'denoised': denoised})
+        dt = sigmas[i + 1] - sigma_hat
+        if sigmas[i + 1] == 0:
+            # Euler method
+            x = x + d * dt
+        else:
+            # Heun's method
+            x_2 = x + d * dt
+            denoised_2 = model(x_2, sigmas[i + 1] * s_in, **extra_args)
+            d_2 = to_d(x_2, sigmas[i + 1], denoised_2)
+            d_prime = (d + d_2) / 2
+            x = x + d_prime * dt
+    return x
+
+
+@torch.no_grad()
+def sample_dpm_2(model, x, sigmas, extra_args=None, callback=None, disable=None, s_churn=0., s_tmin=0., s_tmax=float('inf'), s_noise=1.):
+    """A sampler inspired by DPM-Solver-2 and Algorithm 2 from Karras et al. (2022)."""
+    extra_args = {} if extra_args is None else extra_args
+    s_in = x.new_ones([x.shape[0]])
+    for i in trange(len(sigmas) - 1, disable=disable):
+        gamma = min(s_churn / (len(sigmas) - 1), 2 ** 0.5 - 1) if s_tmin <= sigmas[i] <= s_tmax else 0.
+        eps = torch.randn_like(x) * s_noise
+        sigma_hat = sigmas[i] * (gamma + 1)
+        if gamma > 0:
+            x = x + eps * (sigma_hat ** 2 - sigmas[i] ** 2) ** 0.5
+        denoised = model(x, sigma_hat * s_in, **extra_args)
+        d = to_d(x, sigma_hat, denoised)
+        if callback is not None:
+            callback({'x': x, 'i': i, 'sigma': sigmas[i], 'sigma_hat': sigma_hat, 'denoised': denoised})
+        if sigmas[i + 1] == 0:
+            # Euler method
+            dt = sigmas[i + 1] - sigma_hat
+            x = x + d * dt
+        else:
+            # DPM-Solver-2
+            sigma_mid = sigma_hat.log().lerp(sigmas[i + 1].log(), 0.5).exp()
+            dt_1 = sigma_mid - sigma_hat
+            dt_2 = sigmas[i + 1] - sigma_hat
+            x_2 = x + d * dt_1
+            denoised_2 = model(x_2, sigma_mid * s_in, **extra_args)
+            d_2 = to_d(x_2, sigma_mid, denoised_2)
+            x = x + d_2 * dt_2
+    return x
+
+
+@torch.no_grad()
+def sample_dpm_2_ancestral(model, x, sigmas, extra_args=None, callback=None, disable=None, eta=1., s_noise=1., noise_sampler=None):
+    """Ancestral sampling with DPM-Solver second-order steps."""
+    extra_args = {} if extra_args is None else extra_args
+    noise_sampler = default_noise_sampler(x) if noise_sampler is None else noise_sampler
+    s_in = x.new_ones([x.shape[0]])
+    for i in trange(len(sigmas) - 1, disable=disable):
+        denoised = model(x, sigmas[i] * s_in, **extra_args)
+        sigma_down, sigma_up = get_ancestral_step(sigmas[i], sigmas[i + 1], eta=eta)
+        if callback is not None:
+            callback({'x': x, 'i': i, 'sigma': sigmas[i], 'sigma_hat': sigmas[i], 'denoised': denoised})
+        d = to_d(x, sigmas[i], denoised)
+        if sigma_down == 0:
+            # Euler method
+            dt = sigma_down - sigmas[i]
+            x = x + d * dt
+        else:
+            # DPM-Solver-2
+            sigma_mid = sigmas[i].log().lerp(sigma_down.log(), 0.5).exp()
+            dt_1 = sigma_mid - sigmas[i]
+            dt_2 = sigma_down - sigmas[i]
+            x_2 = x + d * dt_1
+            denoised_2 = model(x_2, sigma_mid * s_in, **extra_args)
+            d_2 = to_d(x_2, sigma_mid, denoised_2)
+            x = x + d_2 * dt_2
+            x = x + noise_sampler(sigmas[i], sigmas[i + 1]) * s_noise * sigma_up
+    return x
+
+
+def linear_multistep_coeff(order, t, i, j):
+    if order - 1 > i:
+        raise ValueError(f'Order {order} too high for step {i}')
+    def fn(tau):
+        prod = 1.
+        for k in range(order):
+            if j == k:
+                continue
+            prod *= (tau - t[i - k]) / (t[i - j] - t[i - k])
+        return prod
+    return integrate.quad(fn, t[i], t[i + 1], epsrel=1e-4)[0]
+
+
+@torch.no_grad()
+def sample_lms(model, x, sigmas, extra_args=None, callback=None, disable=None, order=4):
+    extra_args = {} if extra_args is None else extra_args
+    s_in = x.new_ones([x.shape[0]])
+    sigmas_cpu = sigmas.detach().cpu().numpy()
+    ds = []
+    for i in trange(len(sigmas) - 1, disable=disable):
+        denoised = model(x, sigmas[i] * s_in, **extra_args)
+        d = to_d(x, sigmas[i], denoised)
+        ds.append(d)
+        if len(ds) > order:
+            ds.pop(0)
+        if callback is not None:
+            callback({'x': x, 'i': i, 'sigma': sigmas[i], 'sigma_hat': sigmas[i], 'denoised': denoised})
+        cur_order = min(i + 1, order)
+        coeffs = [linear_multistep_coeff(cur_order, sigmas_cpu, i, j) for j in range(cur_order)]
+        x = x + sum(coeff * d for coeff, d in zip(coeffs, reversed(ds)))
+    return x
+
+
+@torch.no_grad()
+def log_likelihood(model, x, sigma_min, sigma_max, extra_args=None, atol=1e-4, rtol=1e-4):
+    extra_args = {} if extra_args is None else extra_args
+    s_in = x.new_ones([x.shape[0]])
+    v = torch.randint_like(x, 2) * 2 - 1
+    fevals = 0
+    def ode_fn(sigma, x):
+        nonlocal fevals
+        with torch.enable_grad():
+            x = x[0].detach().requires_grad_()
+            denoised = model(x, sigma * s_in, **extra_args)
+            d = to_d(x, sigma, denoised)
+            fevals += 1
+            grad = torch.autograd.grad((d * v).sum(), x)[0]
+            d_ll = (v * grad).flatten(1).sum(1)
+        return d.detach(), d_ll
+    x_min = x, x.new_zeros([x.shape[0]])
+    t = x.new_tensor([sigma_min, sigma_max])
+    sol = odeint(ode_fn, x_min, t, atol=atol, rtol=rtol, method='dopri5')
+    latent, delta_ll = sol[0][-1], sol[1][-1]
+    ll_prior = torch.distributions.Normal(0, sigma_max).log_prob(latent).flatten(1).sum(1)
+    return ll_prior + delta_ll, {'fevals': fevals}
+
+
+class PIDStepSizeController:
+    """A PID controller for ODE adaptive step size control."""
+    def __init__(self, h, pcoeff, icoeff, dcoeff, order=1, accept_safety=0.81, eps=1e-8):
+        self.h = h
+        self.b1 = (pcoeff + icoeff + dcoeff) / order
+        self.b2 = -(pcoeff + 2 * dcoeff) / order
+        self.b3 = dcoeff / order
+        self.accept_safety = accept_safety
+        self.eps = eps
+        self.errs = []
+
+    def limiter(self, x):
+        return 1 + math.atan(x - 1)
+
+    def propose_step(self, error):
+        inv_error = 1 / (float(error) + self.eps)
+        if not self.errs:
+            self.errs = [inv_error, inv_error, inv_error]
+        self.errs[0] = inv_error
+        factor = self.errs[0] ** self.b1 * self.errs[1] ** self.b2 * self.errs[2] ** self.b3
+        factor = self.limiter(factor)
+        accept = factor >= self.accept_safety
+        if accept:
+            self.errs[2] = self.errs[1]
+            self.errs[1] = self.errs[0]
+        self.h *= factor
+        return accept
+
+
+class DPMSolver(nn.Module):
+    """DPM-Solver. See https://arxiv.org/abs/2206.00927."""
+
+    def __init__(self, model, extra_args=None, eps_callback=None, info_callback=None):
+        super().__init__()
+        self.model = model
+        self.extra_args = {} if extra_args is None else extra_args
+        self.eps_callback = eps_callback
+        self.info_callback = info_callback
+
+    def t(self, sigma):
+        return -sigma.log()
+
+    def sigma(self, t):
+        return t.neg().exp()
+
+    def eps(self, eps_cache, key, x, t, *args, **kwargs):
+        if key in eps_cache:
+            return eps_cache[key], eps_cache
+        sigma = self.sigma(t) * x.new_ones([x.shape[0]])
+        eps = (x - self.model(x, sigma, *args, **self.extra_args, **kwargs)) / self.sigma(t)
+        if self.eps_callback is not None:
+            self.eps_callback()
+        return eps, {key: eps, **eps_cache}
+
+    def dpm_solver_1_step(self, x, t, t_next, eps_cache=None):
+        eps_cache = {} if eps_cache is None else eps_cache
+        h = t_next - t
+        eps, eps_cache = self.eps(eps_cache, 'eps', x, t)
+        x_1 = x - self.sigma(t_next) * h.expm1() * eps
+        return x_1, eps_cache
+
+    def dpm_solver_2_step(self, x, t, t_next, r1=1 / 2, eps_cache=None):
+        eps_cache = {} if eps_cache is None else eps_cache
+        h = t_next - t
+        eps, eps_cache = self.eps(eps_cache, 'eps', x, t)
+        s1 = t + r1 * h
+        u1 = x - self.sigma(s1) * (r1 * h).expm1() * eps
+        eps_r1, eps_cache = self.eps(eps_cache, 'eps_r1', u1, s1)
+        x_2 = x - self.sigma(t_next) * h.expm1() * eps - self.sigma(t_next) / (2 * r1) * h.expm1() * (eps_r1 - eps)
+        return x_2, eps_cache
+
+    def dpm_solver_3_step(self, x, t, t_next, r1=1 / 3, r2=2 / 3, eps_cache=None):
+        eps_cache = {} if eps_cache is None else eps_cache
+        h = t_next - t
+        eps, eps_cache = self.eps(eps_cache, 'eps', x, t)
+        s1 = t + r1 * h
+        s2 = t + r2 * h
+        u1 = x - self.sigma(s1) * (r1 * h).expm1() * eps
+        eps_r1, eps_cache = self.eps(eps_cache, 'eps_r1', u1, s1)
+        u2 = x - self.sigma(s2) * (r2 * h).expm1() * eps - self.sigma(s2) * (r2 / r1) * ((r2 * h).expm1() / (r2 * h) - 1) * (eps_r1 - eps)
+        eps_r2, eps_cache = self.eps(eps_cache, 'eps_r2', u2, s2)
+        x_3 = x - self.sigma(t_next) * h.expm1() * eps - self.sigma(t_next) / r2 * (h.expm1() / h - 1) * (eps_r2 - eps)
+        return x_3, eps_cache
+
+    def dpm_solver_fast(self, x, t_start, t_end, nfe, eta=0., s_noise=1., noise_sampler=None):
+        noise_sampler = default_noise_sampler(x) if noise_sampler is None else noise_sampler
+        if not t_end > t_start and eta:
+            raise ValueError('eta must be 0 for reverse sampling')
+
+        m = math.floor(nfe / 3) + 1
+        ts = torch.linspace(t_start, t_end, m + 1, device=x.device)
+
+        if nfe % 3 == 0:
+            orders = [3] * (m - 2) + [2, 1]
+        else:
+            orders = [3] * (m - 1) + [nfe % 3]
+
+        for i in range(len(orders)):
+            eps_cache = {}
+            t, t_next = ts[i], ts[i + 1]
+            if eta:
+                sd, su = get_ancestral_step(self.sigma(t), self.sigma(t_next), eta)
+                t_next_ = torch.minimum(t_end, self.t(sd))
+                su = (self.sigma(t_next) ** 2 - self.sigma(t_next_) ** 2) ** 0.5
+            else:
+                t_next_, su = t_next, 0.
+
+            eps, eps_cache = self.eps(eps_cache, 'eps', x, t)
+            denoised = x - self.sigma(t) * eps
+            if self.info_callback is not None:
+                self.info_callback({'x': x, 'i': i, 't': ts[i], 't_up': t, 'denoised': denoised})
+
+            if orders[i] == 1:
+                x, eps_cache = self.dpm_solver_1_step(x, t, t_next_, eps_cache=eps_cache)
+            elif orders[i] == 2:
+                x, eps_cache = self.dpm_solver_2_step(x, t, t_next_, eps_cache=eps_cache)
+            else:
+                x, eps_cache = self.dpm_solver_3_step(x, t, t_next_, eps_cache=eps_cache)
+
+            x = x + su * s_noise * noise_sampler(self.sigma(t), self.sigma(t_next))
+
+        return x
+
+    def dpm_solver_adaptive(self, x, t_start, t_end, order=3, rtol=0.05, atol=0.0078, h_init=0.05, pcoeff=0., icoeff=1., dcoeff=0., accept_safety=0.81, eta=0., s_noise=1., noise_sampler=None):
+        noise_sampler = default_noise_sampler(x) if noise_sampler is None else noise_sampler
+        if order not in {2, 3}:
+            raise ValueError('order should be 2 or 3')
+        forward = t_end > t_start
+        if not forward and eta:
+            raise ValueError('eta must be 0 for reverse sampling')
+        h_init = abs(h_init) * (1 if forward else -1)
+        atol = torch.tensor(atol)
+        rtol = torch.tensor(rtol)
+        s = t_start
+        x_prev = x
+        accept = True
+        pid = PIDStepSizeController(h_init, pcoeff, icoeff, dcoeff, 1.5 if eta else order, accept_safety)
+        info = {'steps': 0, 'nfe': 0, 'n_accept': 0, 'n_reject': 0}
+
+        while s < t_end - 1e-5 if forward else s > t_end + 1e-5:
+            eps_cache = {}
+            t = torch.minimum(t_end, s + pid.h) if forward else torch.maximum(t_end, s + pid.h)
+            if eta:
+                sd, su = get_ancestral_step(self.sigma(s), self.sigma(t), eta)
+                t_ = torch.minimum(t_end, self.t(sd))
+                su = (self.sigma(t) ** 2 - self.sigma(t_) ** 2) ** 0.5
+            else:
+                t_, su = t, 0.
+
+            eps, eps_cache = self.eps(eps_cache, 'eps', x, s)
+            denoised = x - self.sigma(s) * eps
+
+            if order == 2:
+                x_low, eps_cache = self.dpm_solver_1_step(x, s, t_, eps_cache=eps_cache)
+                x_high, eps_cache = self.dpm_solver_2_step(x, s, t_, eps_cache=eps_cache)
+            else:
+                x_low, eps_cache = self.dpm_solver_2_step(x, s, t_, r1=1 / 3, eps_cache=eps_cache)
+                x_high, eps_cache = self.dpm_solver_3_step(x, s, t_, eps_cache=eps_cache)
+            delta = torch.maximum(atol, rtol * torch.maximum(x_low.abs(), x_prev.abs()))
+            error = torch.linalg.norm((x_low - x_high) / delta) / x.numel() ** 0.5
+            accept = pid.propose_step(error)
+            if accept:
+                x_prev = x_low
+                x = x_high + su * s_noise * noise_sampler(self.sigma(s), self.sigma(t))
+                s = t
+                info['n_accept'] += 1
+            else:
+                info['n_reject'] += 1
+            info['nfe'] += order
+            info['steps'] += 1
+
+            if self.info_callback is not None:
+                self.info_callback({'x': x, 'i': info['steps'] - 1, 't': s, 't_up': s, 'denoised': denoised, 'error': error, 'h': pid.h, **info})
+
+        return x, info
+
+
+@torch.no_grad()
+def sample_dpm_fast(model, x, sigma_min, sigma_max, n, extra_args=None, callback=None, disable=None, eta=0., s_noise=1., noise_sampler=None):
+    """DPM-Solver-Fast (fixed step size). See https://arxiv.org/abs/2206.00927."""
+    if sigma_min <= 0 or sigma_max <= 0:
+        raise ValueError('sigma_min and sigma_max must not be 0')
+    with tqdm(total=n, disable=disable) as pbar:
+        dpm_solver = DPMSolver(model, extra_args, eps_callback=pbar.update)
+        if callback is not None:
+            dpm_solver.info_callback = lambda info: callback({'sigma': dpm_solver.sigma(info['t']), 'sigma_hat': dpm_solver.sigma(info['t_up']), **info})
+        return dpm_solver.dpm_solver_fast(x, dpm_solver.t(torch.tensor(sigma_max)), dpm_solver.t(torch.tensor(sigma_min)), n, eta, s_noise, noise_sampler)
+
+
+@torch.no_grad()
+def sample_dpm_adaptive(model, x, sigma_min, sigma_max, extra_args=None, callback=None, disable=None, order=3, rtol=0.05, atol=0.0078, h_init=0.05, pcoeff=0., icoeff=1., dcoeff=0., accept_safety=0.81, eta=0., s_noise=1., noise_sampler=None, return_info=False):
+    """DPM-Solver-12 and 23 (adaptive step size). See https://arxiv.org/abs/2206.00927."""
+    if sigma_min <= 0 or sigma_max <= 0:
+        raise ValueError('sigma_min and sigma_max must not be 0')
+    with tqdm(disable=disable) as pbar:
+        dpm_solver = DPMSolver(model, extra_args, eps_callback=pbar.update)
+        if callback is not None:
+            dpm_solver.info_callback = lambda info: callback({'sigma': dpm_solver.sigma(info['t']), 'sigma_hat': dpm_solver.sigma(info['t_up']), **info})
+        x, info = dpm_solver.dpm_solver_adaptive(x, dpm_solver.t(torch.tensor(sigma_max)), dpm_solver.t(torch.tensor(sigma_min)), order, rtol, atol, h_init, pcoeff, icoeff, dcoeff, accept_safety, eta, s_noise, noise_sampler)
+    if return_info:
+        return x, info
+    return x
+
+
+@torch.no_grad()
+def sample_dpmpp_2s_ancestral(model, x, sigmas, extra_args=None, callback=None, disable=None, eta=1., s_noise=1., noise_sampler=None):
+    """Ancestral sampling with DPM-Solver++(2S) second-order steps."""
+    extra_args = {} if extra_args is None else extra_args
+    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()
+
+    for i in trange(len(sigmas) - 1, disable=disable):
+        denoised = model(x, sigmas[i] * s_in, **extra_args)
+        sigma_down, sigma_up = get_ancestral_step(sigmas[i], sigmas[i + 1], eta=eta)
+        if callback is not None:
+            callback({'x': x, 'i': i, 'sigma': sigmas[i], 'sigma_hat': sigmas[i], 'denoised': denoised})
+        if sigma_down == 0:
+            # Euler method
+            d = to_d(x, sigmas[i], denoised)
+            dt = sigma_down - sigmas[i]
+            x = x + d * dt
+        else:
+            # DPM-Solver++(2S)
+            t, t_next = t_fn(sigmas[i]), t_fn(sigma_down)
+            r = 1 / 2
+            h = t_next - t
+            s = t + r * h
+            x_2 = (sigma_fn(s) / sigma_fn(t)) * x - (-h * r).expm1() * denoised
+            denoised_2 = model(x_2, sigma_fn(s) * s_in, **extra_args)
+            x = (sigma_fn(t_next) / sigma_fn(t)) * x - (-h).expm1() * denoised_2
+        # Noise addition
+        if sigmas[i + 1] > 0:
+            x = x + noise_sampler(sigmas[i], sigmas[i + 1]) * s_noise * sigma_up
+    return x
+
+
+@torch.no_grad()
+def sample_dpmpp_sde(model, x, sigmas, extra_args=None, callback=None, disable=None, eta=1., s_noise=1., noise_sampler=None, r=1 / 2):
+    """DPM-Solver++ (stochastic)."""
+    sigma_min, sigma_max = sigmas[sigmas > 0].min(), sigmas.max()
+    noise_sampler = BrownianTreeNoiseSampler(x, sigma_min, sigma_max) if noise_sampler is None else noise_sampler
+    extra_args = {} if extra_args is None else extra_args
+    s_in = x.new_ones([x.shape[0]])
+    sigma_fn = lambda t: t.neg().exp()
+    t_fn = lambda sigma: sigma.log().neg()
+
+    for i in trange(len(sigmas) - 1, disable=disable):
+        denoised = model(x, sigmas[i] * s_in, **extra_args)
+        if callback is not None:
+            callback({'x': x, 'i': i, 'sigma': sigmas[i], 'sigma_hat': sigmas[i], 'denoised': denoised})
+        if sigmas[i + 1] == 0:
+            # Euler method
+            d = to_d(x, sigmas[i], denoised)
+            dt = sigmas[i + 1] - sigmas[i]
+            x = x + d * dt
+        else:
+            # DPM-Solver++
+            t, t_next = t_fn(sigmas[i]), t_fn(sigmas[i + 1])
+            h = t_next - t
+            s = t + h * r
+            fac = 1 / (2 * r)
+
+            # Step 1
+            sd, su = get_ancestral_step(sigma_fn(t), sigma_fn(s), eta)
+            s_ = t_fn(sd)
+            x_2 = (sigma_fn(s_) / sigma_fn(t)) * x - (t - s_).expm1() * denoised
+            x_2 = x_2 + noise_sampler(sigma_fn(t), sigma_fn(s)) * s_noise * su
+            denoised_2 = model(x_2, sigma_fn(s) * s_in, **extra_args)
+
+            # Step 2
+            sd, su = get_ancestral_step(sigma_fn(t), sigma_fn(t_next), eta)
+            t_next_ = t_fn(sd)
+            denoised_d = (1 - fac) * denoised + fac * denoised_2
+            x = (sigma_fn(t_next_) / sigma_fn(t)) * x - (t - t_next_).expm1() * denoised_d
+            x = x + noise_sampler(sigma_fn(t), sigma_fn(t_next)) * s_noise * su
+    return x
+
+
+@torch.no_grad()
+def sample_dpmpp_2m(model, x, sigmas, extra_args=None, callback=None, disable=None):
+    """DPM-Solver++(2M)."""
+    extra_args = {} if extra_args is None else extra_args
+    s_in = x.new_ones([x.shape[0]])
+    sigma_fn = lambda t: t.neg().exp()
+    t_fn = lambda sigma: sigma.log().neg()
+    old_denoised = None
+
+    for i in trange(len(sigmas) - 1, disable=disable):
+        denoised = model(x, sigmas[i] * s_in, **extra_args)
+        if callback is not None:
+            callback({'x': x, 'i': i, 'sigma': sigmas[i], 'sigma_hat': sigmas[i], 'denoised': denoised})
+        t, t_next = t_fn(sigmas[i]), t_fn(sigmas[i + 1])
+        h = t_next - t
+        if old_denoised is None or sigmas[i + 1] == 0:
+            x = (sigma_fn(t_next) / sigma_fn(t)) * x - (-h).expm1() * denoised
+        else:
+            h_last = t - t_fn(sigmas[i - 1])
+            r = h_last / h
+            denoised_d = (1 + 1 / (2 * r)) * denoised - (1 / (2 * r)) * old_denoised
+            x = (sigma_fn(t_next) / sigma_fn(t)) * x - (-h).expm1() * denoised_d
+        old_denoised = denoised
+    return x
+
+
+@torch.no_grad()
+def sample_dpmpp_2m_sde(model, x, sigmas, extra_args=None, callback=None, disable=None, eta=1., s_noise=1., noise_sampler=None, solver_type='midpoint'):
+    """DPM-Solver++(2M) SDE."""
+
+    if solver_type not in {'heun', 'midpoint'}:
+        raise ValueError('solver_type must be \'heun\' or \'midpoint\'')
+
+    sigma_min, sigma_max = sigmas[sigmas > 0].min(), sigmas.max()
+    noise_sampler = BrownianTreeNoiseSampler(x, sigma_min, sigma_max) if noise_sampler is None else noise_sampler
+    extra_args = {} if extra_args is None else extra_args
+    s_in = x.new_ones([x.shape[0]])
+
+    old_denoised = None
+    h_last = None
+
+    for i in trange(len(sigmas) - 1, disable=disable):
+        denoised = model(x, sigmas[i] * s_in, **extra_args)
+        if callback is not None:
+            callback({'x': x, 'i': i, 'sigma': sigmas[i], 'sigma_hat': sigmas[i], 'denoised': denoised})
+        if sigmas[i + 1] == 0:
+            # Denoising step
+            x = denoised
+        else:
+            # DPM-Solver++(2M) SDE
+            t, s = -sigmas[i].log(), -sigmas[i + 1].log()
+            h = s - t
+            eta_h = eta * h
+
+            x = sigmas[i + 1] / sigmas[i] * (-eta_h).exp() * x + (-h - eta_h).expm1().neg() * denoised
+
+            if old_denoised is not None:
+                r = h_last / h
+                if solver_type == 'heun':
+                    x = x + ((-h - eta_h).expm1().neg() / (-h - eta_h) + 1) * (1 / r) * (denoised - old_denoised)
+                elif solver_type == 'midpoint':
+                    x = x + 0.5 * (-h - eta_h).expm1().neg() * (1 / r) * (denoised - old_denoised)
+
+            if eta:
+                x = x + noise_sampler(sigmas[i], sigmas[i + 1]) * sigmas[i + 1] * (-2 * eta_h).expm1().neg().sqrt() * s_noise
+
+        old_denoised = denoised
+        h_last = h
+    return x
+
+
+@torch.no_grad()
+def sample_dpmpp_3m_sde(model, x, sigmas, extra_args=None, callback=None, disable=None, eta=1., s_noise=1., noise_sampler=None):
+    """DPM-Solver++(3M) SDE."""
+
+    sigma_min, sigma_max = sigmas[sigmas > 0].min(), sigmas.max()
+    noise_sampler = BrownianTreeNoiseSampler(x, sigma_min, sigma_max) if noise_sampler is None else noise_sampler
+    extra_args = {} if extra_args is None else extra_args
+    s_in = x.new_ones([x.shape[0]])
+
+    denoised_1, denoised_2 = None, None
+    h_1, h_2 = None, None
+
+    for i in trange(len(sigmas) - 1, disable=disable):
+        denoised = model(x, sigmas[i] * s_in, **extra_args)
+        if callback is not None:
+            callback({'x': x, 'i': i, 'sigma': sigmas[i], 'sigma_hat': sigmas[i], 'denoised': denoised})
+        if sigmas[i + 1] == 0:
+            # Denoising step
+            x = denoised
+        else:
+            t, s = -sigmas[i].log(), -sigmas[i + 1].log()
+            h = s - t
+            h_eta = h * (eta + 1)
+
+            x = torch.exp(-h_eta) * x + (-h_eta).expm1().neg() * denoised
+
+            if h_2 is not None:
+                r0 = h_1 / h
+                r1 = h_2 / h
+                d1_0 = (denoised - denoised_1) / r0
+                d1_1 = (denoised_1 - denoised_2) / r1
+                d1 = d1_0 + (d1_0 - d1_1) * r0 / (r0 + r1)
+                d2 = (d1_0 - d1_1) / (r0 + r1)
+                phi_2 = h_eta.neg().expm1() / h_eta + 1
+                phi_3 = phi_2 / h_eta - 0.5
+                x = x + phi_2 * d1 - phi_3 * d2
+            elif h_1 is not None:
+                r = h_1 / h
+                d = (denoised - denoised_1) / r
+                phi_2 = h_eta.neg().expm1() / h_eta + 1
+                x = x + phi_2 * d
+
+            if eta:
+                x = x + noise_sampler(sigmas[i], sigmas[i + 1]) * sigmas[i + 1] * (-2 * h * eta).expm1().neg().sqrt() * s_noise
+
+        denoised_1, denoised_2 = denoised, denoised_1
+        h_1, h_2 = h, h_1
+    return x
+
+@torch.no_grad()
+def sample_dpmpp_2m_alt(model, x, sigmas, extra_args=None, callback=None, disable=None):
+    """DPM-Solver++(2M)."""
+    extra_args = {} if extra_args is None else extra_args
+    s_in = x.new_ones([x.shape[0]])
+    sigma_fn = lambda t: t.neg().exp()
+    t_fn = lambda sigma: sigma.log().neg()
+    old_denoised = None
+
+    for i in trange(len(sigmas) - 1, disable=disable):
+        denoised = model(x, sigmas[i] * s_in, **extra_args)
+        if callback is not None:
+            callback({'x': x, 'i': i, 'sigma': sigmas[i], 'sigma_hat': sigmas[i], 'denoised': denoised})
+        t, t_next = t_fn(sigmas[i]), t_fn(sigmas[i + 1])
+        h = t_next - t
+        
+        t_min = min(sigma_fn(t_next), sigma_fn(t))
+        t_max = max(sigma_fn(t_next), sigma_fn(t))
+
+        if old_denoised is None or sigmas[i + 1] == 0:
+            x = (t_min / t_max) * x - (-h).expm1() * denoised
+        else:
+            h_last = t - t_fn(sigmas[i - 1])
+
+            h_min = min(h_last, h)
+            h_max = max(h_last, h)
+            r = h_max / h_min
+
+            h_d = (h_max + h_min) / 2
+            denoised_d = (1 + 1 / (2 * r)) * denoised - (1 / (2 * r)) * old_denoised
+            x = (t_min / t_max) * x - (-h_d).expm1() * denoised_d
+
+        old_denoised = denoised
+    return x
+
+@torch.no_grad()
+def sample_dpmpp_2m_v1(model, x, sigmas, extra_args=None, callback=None, disable=None):
+    """DPM-Solver++(2M)."""
+    extra_args = {} if extra_args is None else extra_args
+    s_in = x.new_ones([x.shape[0]])
+    sigma_fn = lambda t: t.neg().exp()
+    t_fn = lambda sigma: sigma.log().neg()
+    old_denoised = None
+
+    for i in trange(len(sigmas) - 1, disable=disable):
+        denoised = model(x, sigmas[i] * s_in, **extra_args)
+        if callback is not None:
+            callback({'x': x, 'i': i, 'sigma': sigmas[i], 'sigma_hat': sigmas[i], 'denoised': denoised})
+        t, t_next = t_fn(sigmas[i]), t_fn(sigmas[i + 1])
+        h = t_next - t
+        if old_denoised is None or sigmas[i + 1] == 0:
+            x = (sigma_fn(t_next) / sigma_fn(t)) * x - (-h).expm1() * denoised
+        else:
+            h_last = t - t_fn(sigmas[i - 1])
+            r = h_last / h
+            denoised_d = (1 + 1 / (2 * r)) * denoised - (1 / (2 * r)) * old_denoised
+            x = (sigma_fn(t_next) / sigma_fn(t)) * x - (-h).expm1() * denoised_d
+        sigma_progress = i / len(sigmas)
+        adjustment_factor = 1 + (0.15 * (sigma_progress * sigma_progress))
+        old_denoised = denoised * adjustment_factor
+    return x
+
+@torch.no_grad()
+def sample_dpmpp_2m_test(model, x, sigmas, extra_args=None, callback=None, disable=None):
+    """DPM-Solver++(2M)."""
+    extra_args = {} if extra_args is None else extra_args
+    s_in = x.new_ones([x.shape[0]])
+    sigma_fn = lambda t: t.neg().exp()
+    t_fn = lambda sigma: sigma.log().neg()
+    old_denoised = None
+
+    for i in trange(len(sigmas) - 1, disable=disable):
+        denoised = model(x, sigmas[i] * s_in, **extra_args)
+        if callback is not None:
+            callback({'x': x, 'i': i, 'sigma': sigmas[i], 'sigma_hat': sigmas[i], 'denoised': denoised})
+        t, t_next = t_fn(sigmas[i]), t_fn(sigmas[i + 1])
+        h = t_next - t
+        
+        t_min = min(sigma_fn(t_next), sigma_fn(t))
+        t_max = max(sigma_fn(t_next), sigma_fn(t))
+
+        if old_denoised is None or sigmas[i + 1] == 0:
+            x = (t_min / t_max) * x - (-h).expm1() * denoised
+        else:
+            h_last = t - t_fn(sigmas[i - 1])
+
+            h_min = min(h_last, h)
+            h_max = max(h_last, h)
+            r = h_max / h_min
+
+            h_d = (h_max + h_min) / 2
+            denoised_d = (1 + 1 / (2 * r)) * denoised - (1 / (2 * r)) * old_denoised
+            x = (t_min / t_max) * x - (-h_d).expm1() * denoised_d
+
+        old_denoised = denoised
+    return x
+
+import torch
+from torch import nn
+import torchsde
+from tqdm import trange, tqdm
+from scipy import integrate
+import math
+from modules import shared
+import k_diffusion.sampling
+
+#Citing from K-Diffusion
+#===================================================================================
+def append_zero(x):
+    return torch.cat([x, x.new_zeros([1])])
+
+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)
+
+def get_sigmas_karras(n, sigma_min, sigma_max, rho=7., device='cpu'):
+    """Constructs the noise schedule of Karras et al. (2022)."""
+    ramp = torch.linspace(0, 1, n)
+    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 append_zero(sigmas).to(device)
+
+
+def get_sigmas_exponential(n, sigma_min, sigma_max, device='cpu'):
+    """Constructs an exponential noise schedule."""
+    sigmas = torch.linspace(math.log(sigma_max), math.log(sigma_min), n, device=device).exp()
+    return append_zero(sigmas)
+
+
+def get_sigmas_polyexponential(n, sigma_min, sigma_max, rho=1., device='cpu'):
+    """Constructs an polynomial in log sigma noise schedule."""
+    ramp = torch.linspace(1, 0, n, device=device) ** rho
+    sigmas = torch.exp(ramp * (math.log(sigma_max) - math.log(sigma_min)) + math.log(sigma_min))
+    return append_zero(sigmas)
+
+
+def get_sigmas_vp(n, beta_d=19.9, beta_min=0.1, eps_s=1e-3, device='cpu'):
+    """Constructs a continuous VP noise schedule."""
+    t = torch.linspace(1, eps_s, n, device=device)
+    sigmas = torch.sqrt(torch.exp(beta_d * t ** 2 / 2 + beta_min * t) - 1)
+    return append_zero(sigmas)
+
+def get_ancestral_step(sigma_from, sigma_to, eta=1.):
+    """Calculates the noise level (sigma_down) to step down to and the amount
+    of noise to add (sigma_up) when doing an ancestral sampling step."""
+    if not eta:
+        return sigma_to, 0.
+    sigma_up = min(sigma_to, eta * (sigma_to ** 2 * (sigma_from ** 2 - sigma_to ** 2) / sigma_from ** 2) ** 0.5)
+    sigma_down = (sigma_to ** 2 - sigma_up ** 2) ** 0.5
+    return sigma_down, sigma_up
+
+def default_noise_sampler(x):
+    return lambda sigma, sigma_next: torch.randn_like(x)
+
+def linear_multistep_coeff(order, t, i, j):
+    if order - 1 > i:
+        raise ValueError(f'Order {order} too high for step {i}')
+    def fn(tau):
+        prod = 1.
+        for k in range(order):
+            if j == k:
+                continue
+            prod *= (tau - t[i - k]) / (t[i - j] - t[i - k])
+        return prod
+    return integrate.quad(fn, t[i], t[i + 1], epsrel=1e-4)[0]
+
+class DPMSolver(nn.Module):
+    """DPM-Solver. See https://arxiv.org/abs/2206.00927."""
+
+    def __init__(self, model, extra_args=None, eps_callback=None, info_callback=None):
+        super().__init__()
+        self.model = model
+        self.extra_args = {} if extra_args is None else extra_args
+        self.eps_callback = eps_callback
+        self.info_callback = info_callback
+
+    def t(self, sigma):
+        return -sigma.log()
+
+    def sigma(self, t):
+        return t.neg().exp()
+
+    def eps(self, eps_cache, key, x, t, *args, **kwargs):
+        if key in eps_cache:
+            return eps_cache[key], eps_cache
+        sigma = self.sigma(t) * x.new_ones([x.shape[0]])
+        eps = (x - self.model(x, sigma, *args, **self.extra_args, **kwargs)) / self.sigma(t)
+        if self.eps_callback is not None:
+            self.eps_callback()
+        return eps, {key: eps, **eps_cache}
+
+    def dpm_solver_1_step(self, x, t, t_next, eps_cache=None):
+        eps_cache = {} if eps_cache is None else eps_cache
+        h = t_next - t
+        eps, eps_cache = self.eps(eps_cache, 'eps', x, t)
+        x_1 = x - self.sigma(t_next) * h.expm1() * eps
+        return x_1, eps_cache
+
+    def dpm_solver_2_step(self, x, t, t_next, r1=1 / 2, eps_cache=None):
+        eps_cache = {} if eps_cache is None else eps_cache
+        h = t_next - t
+        eps, eps_cache = self.eps(eps_cache, 'eps', x, t)
+        s1 = t + r1 * h
+        u1 = x - self.sigma(s1) * (r1 * h).expm1() * eps
+        eps_r1, eps_cache = self.eps(eps_cache, 'eps_r1', u1, s1)
+        x_2 = x - self.sigma(t_next) * h.expm1() * eps - self.sigma(t_next) / (2 * r1) * h.expm1() * (eps_r1 - eps)
+        return x_2, eps_cache
+
+    def dpm_solver_3_step(self, x, t, t_next, r1=1 / 3, r2=2 / 3, eps_cache=None):
+        eps_cache = {} if eps_cache is None else eps_cache
+        h = t_next - t
+        eps, eps_cache = self.eps(eps_cache, 'eps', x, t)
+        s1 = t + r1 * h
+        s2 = t + r2 * h
+        u1 = x - self.sigma(s1) * (r1 * h).expm1() * eps
+        eps_r1, eps_cache = self.eps(eps_cache, 'eps_r1', u1, s1)
+        u2 = x - self.sigma(s2) * (r2 * h).expm1() * eps - self.sigma(s2) * (r2 / r1) * ((r2 * h).expm1() / (r2 * h) - 1) * (eps_r1 - eps)
+        eps_r2, eps_cache = self.eps(eps_cache, 'eps_r2', u2, s2)
+        x_3 = x - self.sigma(t_next) * h.expm1() * eps - self.sigma(t_next) / r2 * (h.expm1() / h - 1) * (eps_r2 - eps)
+        return x_3, eps_cache
+
+    def dpm_solver_fast(self, x, t_start, t_end, nfe, eta=0., s_noise=1., noise_sampler=None):
+        noise_sampler = default_noise_sampler(x) if noise_sampler is None else noise_sampler
+        if not t_end > t_start and eta:
+            raise ValueError('eta must be 0 for reverse sampling')
+
+        m = math.floor(nfe / 3) + 1
+        ts = torch.linspace(t_start, t_end, m + 1, device=x.device)
+
+        if nfe % 3 == 0:
+            orders = [3] * (m - 2) + [2, 1]
+        else:
+            orders = [3] * (m - 1) + [nfe % 3]
+
+        for i in range(len(orders)):
+            eps_cache = {}
+            t, t_next = ts[i], ts[i + 1]
+            if eta:
+                sd, su = get_ancestral_step(self.sigma(t), self.sigma(t_next), eta)
+                t_next_ = torch.minimum(t_end, self.t(sd))
+                su = (self.sigma(t_next) ** 2 - self.sigma(t_next_) ** 2) ** 0.5
+            else:
+                t_next_, su = t_next, 0.
+
+            eps, eps_cache = self.eps(eps_cache, 'eps', x, t)
+            denoised = x - self.sigma(t) * eps
+            if self.info_callback is not None:
+                self.info_callback({'x': x, 'i': i, 't': ts[i], 't_up': t, 'denoised': denoised})
+
+            if orders[i] == 1:
+                x, eps_cache = self.dpm_solver_1_step(x, t, t_next_, eps_cache=eps_cache)
+            elif orders[i] == 2:
+                x, eps_cache = self.dpm_solver_2_step(x, t, t_next_, eps_cache=eps_cache)
+            else:
+                x, eps_cache = self.dpm_solver_3_step(x, t, t_next_, eps_cache=eps_cache)
+
+            x = x + su * s_noise * noise_sampler(self.sigma(t), self.sigma(t_next))
+
+        return x
+
+    def dpm_solver_adaptive(self, x, t_start, t_end, order=3, rtol=0.05, atol=0.0078, h_init=0.05, pcoeff=0., icoeff=1.,
+                            dcoeff=0., accept_safety=0.81, eta=0., s_noise=1., noise_sampler=None):
+        noise_sampler = default_noise_sampler(x) if noise_sampler is None else noise_sampler
+        if order not in {2, 3}:
+            raise ValueError('order should be 2 or 3')
+        forward = t_end > t_start
+        if not forward and eta:
+            raise ValueError('eta must be 0 for reverse sampling')
+        h_init = abs(h_init) * (1 if forward else -1)
+        atol = torch.tensor(atol)
+        rtol = torch.tensor(rtol)
+        s = t_start
+        x_prev = x
+        accept = True
+        pid = PIDStepSizeController(h_init, pcoeff, icoeff, dcoeff, 1.5 if eta else order, accept_safety)
+        info = {'steps': 0, 'nfe': 0, 'n_accept': 0, 'n_reject': 0}
+
+        while s < t_end - 1e-5 if forward else s > t_end + 1e-5:
+            eps_cache = {}
+            t = torch.minimum(t_end, s + pid.h) if forward else torch.maximum(t_end, s + pid.h)
+            if eta:
+                sd, su = get_ancestral_step(self.sigma(s), self.sigma(t), eta)
+                t_ = torch.minimum(t_end, self.t(sd))
+                su = (self.sigma(t) ** 2 - self.sigma(t_) ** 2) ** 0.5
+            else:
+                t_, su = t, 0.
+
+            eps, eps_cache = self.eps(eps_cache, 'eps', x, s)
+            denoised = x - self.sigma(s) * eps
+
+            if order == 2:
+                x_low, eps_cache = self.dpm_solver_1_step(x, s, t_, eps_cache=eps_cache)
+                x_high, eps_cache = self.dpm_solver_2_step(x, s, t_, eps_cache=eps_cache)
+            else:
+                x_low, eps_cache = self.dpm_solver_2_step(x, s, t_, r1=1 / 3, eps_cache=eps_cache)
+                x_high, eps_cache = self.dpm_solver_3_step(x, s, t_, eps_cache=eps_cache)
+            delta = torch.maximum(atol, rtol * torch.maximum(x_low.abs(), x_prev.abs()))
+            error = torch.linalg.norm((x_low - x_high) / delta) / x.numel() ** 0.5
+            accept = pid.propose_step(error)
+            if accept:
+                x_prev = x_low
+                x = x_high + su * s_noise * noise_sampler(self.sigma(s), self.sigma(t))
+                s = t
+                info['n_accept'] += 1
+            else:
+                info['n_reject'] += 1
+            info['nfe'] += order
+            info['steps'] += 1
+
+            if self.info_callback is not None:
+                self.info_callback(
+                    {'x': x, 'i': info['steps'] - 1, 't': s, 't_up': s, 'denoised': denoised, 'error': error,
+                     'h': pid.h, **info})
+
+        return x, info
+
+class BatchedBrownianTree:
+    """A wrapper around torchsde.BrownianTree that enables batches of entropy."""
+
+    def __init__(self, x, t0, t1, seed=None, **kwargs):
+        t0, t1, self.sign = self.sort(t0, t1)
+        w0 = kwargs.get('w0', torch.zeros_like(x))
+        if seed is None:
+            seed = torch.randint(0, 2 ** 63 - 1, []).item()
+        self.batched = True
+        try:
+            assert len(seed) == x.shape[0]
+            w0 = w0[0]
+        except TypeError:
+            seed = [seed]
+            self.batched = False
+        self.trees = [torchsde.BrownianTree(t0, w0, t1, entropy=s, **kwargs) for s in seed]
+
+    @staticmethod
+    def sort(a, b):
+        return (a, b, 1) if a < b else (b, a, -1)
+
+    def __call__(self, t0, t1):
+        t0, t1, sign = self.sort(t0, t1)
+        w = torch.stack([tree(t0, t1) for tree in self.trees]) * (self.sign * sign)
+        return w if self.batched else w[0]
+
+class BrownianTreeNoiseSampler:
+    """A noise sampler backed by a torchsde.BrownianTree.
+
+    Args:
+        x (Tensor): The tensor whose shape, device and dtype to use to generate
+            random samples.
+        sigma_min (float): The low end of the valid interval.
+        sigma_max (float): The high end of the valid interval.
+        seed (int or List[int]): The random seed. If a list of seeds is
+            supplied instead of a single integer, then the noise sampler will
+            use one BrownianTree per batch item, each with its own seed.
+        transform (callable): A function that maps sigma to the sampler's
+            internal timestep.
+    """
+
+    def __init__(self, x, sigma_min, sigma_max, seed=None, transform=lambda x: x):
+        self.transform = transform
+        t0, t1 = self.transform(torch.as_tensor(sigma_min)), self.transform(torch.as_tensor(sigma_max))
+        self.tree = BatchedBrownianTree(x, t0, t1, seed)
+
+    def __call__(self, sigma, sigma_next):
+        t0, t1 = self.transform(torch.as_tensor(sigma)), self.transform(torch.as_tensor(sigma_next))
+        return self.tree(t0, t1) / (t1 - t0).abs().sqrt()
+
+#===================================================================================
+
+@torch.no_grad()
+def sample_skip(model, x, sigmas, extra_args=None, callback=None, disable=None, s_churn=0., s_tmin=0., s_tmax=float('inf'), s_noise=1.):
+    """Implements Algorithm 2 (Euler steps) from Karras et al. (2022)."""
+    extra_args = {} if extra_args is None else extra_args
+    return x
+
+@torch.no_grad()
+def sample_euler(model, x, sigmas, extra_args=None, callback=None, disable=None, s_churn=0., s_tmin=0., s_tmax=float('inf'), s_noise=1.):
+    """Implements Algorithm 2 (Euler steps) from Karras et al. (2022)."""
+    extra_args = {} if extra_args is None else extra_args
+    s_in = x.new_ones([x.shape[0]])
+    for i in trange(len(sigmas) - 1, disable=disable):
+        gamma = min(s_churn / (len(sigmas) - 1), 2 ** 0.5 - 1) if s_tmin <= sigmas[i] <= s_tmax else 0.
+        eps = torch.randn_like(x) * s_noise
+        sigma_hat = sigmas[i] * (gamma + 1)
+        if gamma > 0:
+            x = x + eps * (sigma_hat ** 2 - sigmas[i] ** 2) ** 0.5
+        denoised = model(x, sigma_hat * s_in, **extra_args)
+        d = to_d(x, sigma_hat, denoised)
+        if callback is not None:
+            callback({'x': x, 'i': i, 'sigma': sigmas[i], 'sigma_hat': sigma_hat, 'denoised': denoised})
+        dt = sigmas[i + 1] - sigma_hat
+        # Euler method
+        x = x + d * dt
+    return x
+
+@torch.no_grad()
+def sample_euler_ancestral(model, x, sigmas, extra_args=None, callback=None, disable=None, eta=1., s_noise=1., noise_sampler=None):
+    """Ancestral sampling with Euler method steps."""
+    extra_args = {} if extra_args is None else extra_args
+    noise_sampler = default_noise_sampler(x) if noise_sampler is None else noise_sampler
+    s_in = x.new_ones([x.shape[0]])
+    for i in trange(len(sigmas) - 1, disable=disable):
+        denoised = model(x, sigmas[i] * s_in, **extra_args)
+        sigma_down, sigma_up = get_ancestral_step(sigmas[i], sigmas[i + 1], eta=eta)
+        if callback is not None:
+            callback({'x': x, 'i': i, 'sigma': sigmas[i], 'sigma_hat': sigmas[i], 'denoised': denoised})
+        d = to_d(x, sigmas[i], denoised)
+        # Euler method
+        dt = sigma_down - sigmas[i]
+        x = x + d * dt
+        if sigmas[i + 1] > 0:
+            x = x + noise_sampler(sigmas[i], sigmas[i + 1]) * s_noise * sigma_up
+    return x
+
+@torch.no_grad()
+def sample_heun(model, x, sigmas, extra_args=None, callback=None, disable=None, s_churn=0., s_tmin=0., s_tmax=float('inf'), s_noise=1.):
+    """Implements Algorithm 2 (Heun steps) from Karras et al. (2022)."""
+    extra_args = {} if extra_args is None else extra_args
+    s_in = x.new_ones([x.shape[0]])
+    s_end = sigmas[-1]
+    for i in trange(len(sigmas) - 1, disable=disable):
+        gamma = min(s_churn / (len(sigmas) - 1), 2 ** 0.5 - 1) if s_tmin <= sigmas[i] <= s_tmax else 0.
+        eps = torch.randn_like(x) * s_noise
+        sigma_hat = sigmas[i] * (gamma + 1)
+        if gamma > 0:
+            x = x + eps * (sigma_hat ** 2 - sigmas[i] ** 2) ** 0.5
+        denoised = model(x, sigma_hat * s_in, **extra_args)
+        d = to_d(x, sigma_hat, denoised)
+        if callback is not None:
+            callback({'x': x, 'i': i, 'sigma': sigmas[i], 'sigma_hat': sigma_hat, 'denoised': denoised})
+        dt = sigmas[i + 1] - sigma_hat
+        if sigmas[i + 1] == s_end:
+            # Euler method
+            x = x + d * dt
+        else:
+            # Heun's method
+            x_2 = x + d * dt
+            denoised_2 = model(x_2, sigmas[i + 1] * s_in, **extra_args)
+            d_2 = to_d(x_2, sigmas[i + 1], denoised_2)
+            d_prime = (d + d_2) / 2
+            x = x + d_prime * dt
+
+    return x
+
+@torch.no_grad()
+def sample_heunpp2(model, x, sigmas, extra_args=None, callback=None, disable=None, s_churn=0., s_tmin=0., s_tmax=float('inf'), s_noise=1.):
+    """Implements Algorithm 2 (Heun steps) from Karras et al. (2022)."""
+    extra_args = {} if extra_args is None else extra_args
+    s_in = x.new_ones([x.shape[0]])
+    s_end = sigmas[-1]
+    for i in trange(len(sigmas) - 1, disable=disable):
+        gamma = min(s_churn / (len(sigmas) - 1), 2 ** 0.5 - 1) if s_tmin <= sigmas[i] <= s_tmax else 0.
+        eps = torch.randn_like(x) * s_noise
+        sigma_hat = sigmas[i] * (gamma + 1)
+        if gamma > 0:
+            x = x + eps * (sigma_hat ** 2 - sigmas[i] ** 2) ** 0.5
+        denoised = model(x, sigma_hat * s_in, **extra_args)
+        d = to_d(x, sigma_hat, denoised)
+        if callback is not None:
+            callback({'x': x, 'i': i, 'sigma': sigmas[i], 'sigma_hat': sigma_hat, 'denoised': denoised})
+        dt = sigmas[i + 1] - sigma_hat
+        if sigmas[i + 1] == s_end:
+            # Euler method
+            x = x + d * dt
+        elif sigmas[i + 2] == s_end:
+
+            # Heun's method
+            x_2 = x + d * dt
+            denoised_2 = model(x_2, sigmas[i + 1] * s_in, **extra_args)
+            d_2 = to_d(x_2, sigmas[i + 1], denoised_2)
+
+            w = 2 * sigmas[0]
+            w2 = sigmas[i+1]/w
+            w1 = 1 - w2
+
+            d_prime = d * w1 + d_2 * w2
+
+
+            x = x + d_prime * dt
+
+        else:
+            # Heun++
+            x_2 = x + d * dt
+            denoised_2 = model(x_2, sigmas[i + 1] * s_in, **extra_args)
+            d_2 = to_d(x_2, sigmas[i + 1], denoised_2)
+            dt_2 = sigmas[i + 2] - sigmas[i + 1]
+
+            x_3 = x_2 + d_2 * dt_2
+            denoised_3 = model(x_3, sigmas[i + 2] * s_in, **extra_args)
+            d_3 = to_d(x_3, sigmas[i + 2], denoised_3)
+
+            w = 3 * sigmas[0]
+            w2 = sigmas[i + 1] / w
+            w3 = sigmas[i + 2] / w
+            w1 = 1 - w2 - w3
+
+            d_prime = w1 * d + w2 * d_2 + w3 * d_3
+            x = x + d_prime * dt
+    return x
+
+@torch.no_grad()
+def sample_lms(model, x, sigmas, extra_args=None, callback=None, disable=None, order=4):
+    extra_args = {} if extra_args is None else extra_args
+    s_in = x.new_ones([x.shape[0]])
+    sigmas_cpu = sigmas.detach().cpu().numpy()
+    ds = []
+    for i in trange(len(sigmas) - 1, disable=disable):
+        denoised = model(x, sigmas[i] * s_in, **extra_args)
+        d = to_d(x, sigmas[i], denoised)
+        ds.append(d)
+        if len(ds) > order:
+            ds.pop(0)
+        if callback is not None:
+            callback({'x': x, 'i': i, 'sigma': sigmas[i], 'sigma_hat': sigmas[i], 'denoised': denoised})
+        cur_order = min(i + 1, order)
+        coeffs = [linear_multistep_coeff(cur_order, sigmas_cpu, i, j) for j in range(cur_order)]
+        x = x + sum(coeff * d for coeff, d in zip(coeffs, reversed(ds)))
+
+    return x
+
+@torch.no_grad()
+def sample_dpm_2(model, x, sigmas, extra_args=None, callback=None, disable=None, s_churn=0., s_tmin=0., s_tmax=float('inf'), s_noise=1.):
+    """A sampler inspired by DPM-Solver-2 and Algorithm 2 from Karras et al. (2022)."""
+    extra_args = {} if extra_args is None else extra_args
+    s_in = x.new_ones([x.shape[0]])
+    s_end = sigmas[-1]
+    for i in trange(len(sigmas) - 1, disable=disable):
+        gamma = min(s_churn / (len(sigmas) - 1), 2 ** 0.5 - 1) if s_tmin <= sigmas[i] <= s_tmax else 0.
+        eps = torch.randn_like(x) * s_noise
+        sigma_hat = sigmas[i] * (gamma + 1)
+        if gamma > 0:
+            x = x + eps * (sigma_hat ** 2 - sigmas[i] ** 2) ** 0.5
+        denoised = model(x, sigma_hat * s_in, **extra_args)
+        d = to_d(x, sigma_hat, denoised)
+        if callback is not None:
+            callback({'x': x, 'i': i, 'sigma': sigmas[i], 'sigma_hat': sigma_hat, 'denoised': denoised})
+        if sigmas[i + 1] == s_end:
+            # Euler method
+            dt = sigmas[i + 1] - sigma_hat
+            x = x + d * dt
+        else:
+            # DPM-Solver-2
+            sigma_mid = sigma_hat.log().lerp(sigmas[i + 1].log(), 0.5).exp()
+            dt_1 = sigma_mid - sigma_hat
+            dt_2 = sigmas[i + 1] - sigma_hat
+            x_2 = x + d * dt_1
+            denoised_2 = model(x_2, sigma_mid * s_in, **extra_args)
+            d_2 = to_d(x_2, sigma_mid, denoised_2)
+            x = x + d_2 * dt_2
+    return x
+
+@torch.no_grad()
+def sample_dpm_2_ancestral(model, x, sigmas, extra_args=None, callback=None, disable=None, eta=1., s_noise=1., noise_sampler=None):
+    """Ancestral sampling with DPM-Solver second-order steps."""
+    extra_args = {} if extra_args is None else extra_args
+    noise_sampler = default_noise_sampler(x) if noise_sampler is None else noise_sampler
+    s_in = x.new_ones([x.shape[0]])
+    s_end = sigmas[-1]
+    for i in trange(len(sigmas) - 1, disable=disable):
+        denoised = model(x, sigmas[i] * s_in, **extra_args)
+        sigma_down, sigma_up = get_ancestral_step(sigmas[i], sigmas[i + 1], eta=eta)
+        if callback is not None:
+            callback({'x': x, 'i': i, 'sigma': sigmas[i], 'sigma_hat': sigmas[i], 'denoised': denoised})
+        d = to_d(x, sigmas[i], denoised)
+        if sigma_down == s_end:
+            # Euler method
+            dt = sigma_down - sigmas[i]
+            x = x + d * dt
+        else:
+            # DPM-Solver-2
+            sigma_mid = sigmas[i].log().lerp(sigma_down.log(), 0.5).exp()
+            dt_1 = sigma_mid - sigmas[i]
+            dt_2 = sigma_down - sigmas[i]
+            x_2 = x + d * dt_1
+            denoised_2 = model(x_2, sigma_mid * s_in, **extra_args)
+            d_2 = to_d(x_2, sigma_mid, denoised_2)
+            x = x + d_2 * dt_2
+            x = x + noise_sampler(sigmas[i], sigmas[i + 1]) * s_noise * sigma_up
+    return x
+
+@torch.no_grad()
+def sample_dpmpp_2s_ancestral(model, x, sigmas, extra_args=None, callback=None, disable=None, eta=1., s_noise=1., noise_sampler=None):
+    """Ancestral sampling with DPM-Solver++(2S) second-order steps."""
+    extra_args = {} if extra_args is None else extra_args
+    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()
+    s_end = sigmas[-1]
+
+    for i in trange(len(sigmas) - 1, disable=disable):
+        denoised = model(x, sigmas[i] * s_in, **extra_args)
+        sigma_down, sigma_up = get_ancestral_step(sigmas[i], sigmas[i + 1], eta=eta)
+        if callback is not None:
+            callback({'x': x, 'i': i, 'sigma': sigmas[i], 'sigma_hat': sigmas[i], 'denoised': denoised})
+        if sigma_down == s_end:
+            # Euler method
+            d = to_d(x, sigmas[i], denoised)
+            dt = sigma_down - sigmas[i]
+            x = x + d * dt
+        else:
+            # DPM-Solver++(2S)
+            t, t_next = t_fn(sigmas[i]), t_fn(sigma_down)
+            r = 1 / 2
+            h = t_next - t
+            s = t + r * h
+            x_2 = (sigma_fn(s) / sigma_fn(t)) * x - (-h * r).expm1() * denoised
+            denoised_2 = model(x_2, sigma_fn(s) * s_in, **extra_args)
+            x = (sigma_fn(t_next) / sigma_fn(t)) * x - (-h).expm1() * denoised_2
+        # Noise addition
+        if sigmas[i + 1] > 0:
+            x = x + noise_sampler(sigmas[i], sigmas[i + 1]) * s_noise * sigma_up
+    return x
+
+@torch.no_grad()
+def sample_dpmpp_sde(model, x, sigmas, extra_args=None, callback=None, disable=None, eta=1., s_noise=1., noise_sampler=None, r=1 / 2):
+    """DPM-Solver++ (stochastic)."""
+    sigma_min, sigma_max = sigmas[sigmas > 0].min(), sigmas.max()
+    noise_sampler = BrownianTreeNoiseSampler(x, sigma_min, sigma_max) if noise_sampler is None else noise_sampler
+    extra_args = {} if extra_args is None else extra_args
+    s_in = x.new_ones([x.shape[0]])
+    sigma_fn = lambda t: t.neg().exp()
+    t_fn = lambda sigma: sigma.log().neg()
+    s_end = sigmas[-1]
+
+    for i in trange(len(sigmas) - 1, disable=disable):
+        denoised = model(x, sigmas[i] * s_in, **extra_args)
+        if callback is not None:
+            callback({'x': x, 'i': i, 'sigma': sigmas[i], 'sigma_hat': sigmas[i], 'denoised': denoised})
+        if sigmas[i + 1] == s_end:
+            # Euler method
+            d = to_d(x, sigmas[i], denoised)
+            dt = sigmas[i + 1] - sigmas[i]
+            x = x + d * dt
+        else:
+            # DPM-Solver++
+            t, t_next = t_fn(sigmas[i]), t_fn(sigmas[i + 1])
+            h = t_next - t
+            s = t + h * r
+            fac = 1 / (2 * r)
+
+            # Step 1
+            sd, su = get_ancestral_step(sigma_fn(t), sigma_fn(s), eta)
+            s_ = t_fn(sd)
+            x_2 = (sigma_fn(s_) / sigma_fn(t)) * x - (t - s_).expm1() * denoised
+            x_2 = x_2 + noise_sampler(sigma_fn(t), sigma_fn(s)) * s_noise * su
+            denoised_2 = model(x_2, sigma_fn(s) * s_in, **extra_args)
+
+            # Step 2
+            sd, su = get_ancestral_step(sigma_fn(t), sigma_fn(t_next), eta)
+            t_next_ = t_fn(sd)
+            denoised_d = (1 - fac) * denoised + fac * denoised_2
+            x = (sigma_fn(t_next_) / sigma_fn(t)) * x - (t - t_next_).expm1() * denoised_d
+            x = x + noise_sampler(sigma_fn(t), sigma_fn(t_next)) * s_noise * su
+    return x
+
+@torch.no_grad()
+def sample_dpmpp_2m(model, x, sigmas, extra_args=None, callback=None, disable=None):
+    """DPM-Solver++(2M)."""
+    extra_args = {} if extra_args is None else extra_args
+    s_in = x.new_ones([x.shape[0]])
+    sigma_fn = lambda t: t.neg().exp()
+    t_fn = lambda sigma: sigma.log().neg()
+    old_denoised = None
+    s_end = sigmas[-1]
+
+    for i in trange(len(sigmas) - 1, disable=disable):
+        denoised = model(x, sigmas[i] * s_in, **extra_args)
+        if callback is not None:
+            callback({'x': x, 'i': i, 'sigma': sigmas[i], 'sigma_hat': sigmas[i], 'denoised': denoised})
+        t, t_next = t_fn(sigmas[i]), t_fn(sigmas[i + 1])
+        h = t_next - t
+        if old_denoised is None or sigmas[i + 1] == s_end:
+            x = (sigma_fn(t_next) / sigma_fn(t)) * x - (-h).expm1() * denoised
+        else:
+            h_last = t - t_fn(sigmas[i - 1])
+            r = h_last / h
+            denoised_d = (1 + 1 / (2 * r)) * denoised - (1 / (2 * r)) * old_denoised
+            x = (sigma_fn(t_next) / sigma_fn(t)) * x - (-h).expm1() * denoised_d
+        old_denoised = denoised
+    return x
+
+@torch.no_grad()
+def sample_dpmpp_2m_sde(model, x, sigmas, extra_args=None, callback=None, disable=None, eta=1., s_noise=1., noise_sampler=None, solver_type='midpoint'):
+    """DPM-Solver++(2M) SDE."""
+
+    if solver_type not in {'heun', 'midpoint'}:
+        raise ValueError('solver_type must be \'heun\' or \'midpoint\'')
+
+    sigma_min, sigma_max = sigmas[sigmas > 0].min(), sigmas.max()
+    noise_sampler = BrownianTreeNoiseSampler(x, sigma_min, sigma_max) if noise_sampler is None else noise_sampler
+    extra_args = {} if extra_args is None else extra_args
+    s_in = x.new_ones([x.shape[0]])
+    s_end = sigmas[-1]
+
+    old_denoised = None
+    h_last = None
+
+    for i in trange(len(sigmas) - 1, disable=disable):
+        denoised = model(x, sigmas[i] * s_in, **extra_args)
+        if callback is not None:
+            callback({'x': x, 'i': i, 'sigma': sigmas[i], 'sigma_hat': sigmas[i], 'denoised': denoised})
+        if sigmas[i + 1] == s_end:
+            # Denoising step
+            x = denoised
+        else:
+            # DPM-Solver++(2M) SDE
+            t, s = -sigmas[i].log(), -sigmas[i + 1].log()
+            h = s - t
+            eta_h = eta * h
+
+            x = sigmas[i + 1] / sigmas[i] * (-eta_h).exp() * x + (-h - eta_h).expm1().neg() * denoised
+
+            if old_denoised is not None:
+                r = h_last / h
+                if solver_type == 'heun':
+                    x = x + ((-h - eta_h).expm1().neg() / (-h - eta_h) + 1) * (1 / r) * (denoised - old_denoised)
+                elif solver_type == 'midpoint':
+                    x = x + 0.5 * (-h - eta_h).expm1().neg() * (1 / r) * (denoised - old_denoised)
+
+            if eta:
+                x = x + noise_sampler(sigmas[i], sigmas[i + 1]) * sigmas[i + 1] * (-2 * eta_h).expm1().neg().sqrt() * s_noise
+
+        old_denoised = denoised
+        h_last = h
+    return x
+
+@torch.no_grad()
+def sample_dpmpp_3m_sde(model, x, sigmas, extra_args=None, callback=None, disable=None, eta=1., s_noise=1., noise_sampler=None):
+    """DPM-Solver++(3M) SDE."""
+
+    sigma_min, sigma_max = sigmas[sigmas > 0].min(), sigmas.max()
+    noise_sampler = BrownianTreeNoiseSampler(x, sigma_min, sigma_max) if noise_sampler is None else noise_sampler
+    extra_args = {} if extra_args is None else extra_args
+    s_in = x.new_ones([x.shape[0]])
+    s_end = sigmas[-1]
+
+    denoised_1, denoised_2 = None, None
+    h_1, h_2 = None, None
+
+    for i in trange(len(sigmas) - 1, disable=disable):
+        denoised = model(x, sigmas[i] * s_in, **extra_args)
+        if callback is not None:
+            callback({'x': x, 'i': i, 'sigma': sigmas[i], 'sigma_hat': sigmas[i], 'denoised': denoised})
+        if sigmas[i + 1] == s_end:
+            # Denoising step
+            x = denoised
+        else:
+            t, s = -sigmas[i].log(), -sigmas[i + 1].log()
+            h = s - t
+            h_eta = h * (eta + 1)
+
+            x = torch.exp(-h_eta) * x + (-h_eta).expm1().neg() * denoised
+
+            if h_2 is not None:
+                r0 = h_1 / h
+                r1 = h_2 / h
+                d1_0 = (denoised - denoised_1) / r0
+                d1_1 = (denoised_1 - denoised_2) / r1
+                d1 = d1_0 + (d1_0 - d1_1) * r0 / (r0 + r1)
+                d2 = (d1_0 - d1_1) / (r0 + r1)
+                phi_2 = h_eta.neg().expm1() / h_eta + 1
+                phi_3 = phi_2 / h_eta - 0.5
+                x = x + phi_2 * d1 - phi_3 * d2
+            elif h_1 is not None:
+                r = h_1 / h
+                d = (denoised - denoised_1) / r
+                phi_2 = h_eta.neg().expm1() / h_eta + 1
+                x = x + phi_2 * d
+
+            if eta:
+                x = x + noise_sampler(sigmas[i], sigmas[i + 1]) * sigmas[i + 1] * (-2 * h * eta).expm1().neg().sqrt() * s_noise
+
+        denoised_1, denoised_2 = denoised, denoised_1
+        h_1, h_2 = h, h_1
+    return x
+
+@torch.no_grad()
+def lcm_sampler(model, x, sigmas, extra_args=None, callback=None, disable=None, noise_sampler=None):
+    extra_args = {} if extra_args is None else extra_args
+    noise_sampler = default_noise_sampler(x) if noise_sampler is None else noise_sampler
+    s_end = sigmas[-1]
+    s_in = x.new_ones([x.shape[0]])
+    for i in trange(len(sigmas) - 1, disable=disable):
+        denoised = model(x, sigmas[i] * s_in, **extra_args)
+        if callback is not None:
+            callback({'x': x, 'i': i, 'sigma': sigmas[i], 'sigma_hat': sigmas[i], 'denoised': denoised})
+
+        x = denoised
+        if sigmas[i + 1] > s_end:
+            x += sigmas[i + 1] * noise_sampler(sigmas[i], sigmas[i + 1])
+    return x
+
+@torch.no_grad()
+def restart_sampler(model, x, sigmas, extra_args=None, callback=None, disable=None, s_noise=1., restart_list=None):
+    """Implements restart sampling in Restart Sampling for Improving Generative Processes (2023)
+    Restart_list format: {min_sigma: [ restart_steps, restart_times, max_sigma]}
+    If restart_list is None: will choose restart_list automatically, otherwise will use the given restart_list
+    """
+    extra_args = {} if extra_args is None else extra_args
+    s_in = x.new_ones([x.shape[0]])
+    step_id = 0
+    s_end = sigmas[-1]
+
+    def heun_step(x, old_sigma, new_sigma, second_order=True):
+        nonlocal step_id, s_end
+        denoised = model(x, old_sigma * s_in, **extra_args)
+        d = to_d(x, old_sigma, denoised)
+        if callback is not None:
+            callback({'x': x, 'i': step_id, 'sigma': new_sigma, 'sigma_hat': old_sigma, 'denoised': denoised})
+        dt = new_sigma - old_sigma
+        if new_sigma == s_end or not second_order:
+            # Euler method
+            x = x + d * dt
+        else:
+            # Heun's method
+            x_2 = x + d * dt
+            denoised_2 = model(x_2, new_sigma * s_in, **extra_args)
+            d_2 = to_d(x_2, new_sigma, denoised_2)
+            d_prime = (d + d_2) / 2
+            x = x + d_prime * dt
+        step_id += 1
+        return x
+
+    steps = sigmas.shape[0] - 1
+    if restart_list is None:
+        if steps >= 20:
+            restart_steps = 9
+            restart_times = 1
+            if steps >= 36:
+                restart_steps = steps // 4
+                restart_times = 2
+            sigmas = get_sigmas_karras(steps - restart_steps * restart_times, sigmas[-2].item(), sigmas[0].item(), device=sigmas.device)
+            restart_list = {0.1: [restart_steps + 1, restart_times, 2]}
+        else:
+            restart_list = {}
+
+    restart_list = {int(torch.argmin(abs(sigmas - key), dim=0)): value for key, value in restart_list.items()}
+
+    step_list = []
+    for i in range(len(sigmas) - 1):
+        step_list.append((sigmas[i], sigmas[i + 1]))
+        if i + 1 in restart_list:
+            restart_steps, restart_times, restart_max = restart_list[i + 1]
+            min_idx = i + 1
+            max_idx = int(torch.argmin(abs(sigmas - restart_max), dim=0))
+            if max_idx < min_idx:
+                sigma_restart = get_sigmas_karras(restart_steps, sigmas[min_idx].item(), sigmas[max_idx].item(), device=sigmas.device)[:-1]
+                while restart_times > 0:
+                    restart_times -= 1
+                    step_list.extend([(old_sigma, new_sigma) for (old_sigma, new_sigma) in zip(sigma_restart[:-1], sigma_restart[1:])])
+
+    last_sigma = None
+    for old_sigma, new_sigma in tqdm(step_list, disable=disable):
+        if last_sigma is None:
+            last_sigma = old_sigma
+        elif last_sigma < old_sigma:
+            x = x + k_diffusion.sampling.torch.randn_like(x) * s_noise * (old_sigma ** 2 - last_sigma ** 2) ** 0.5
+        x = heun_step(x, old_sigma, new_sigma)
+        last_sigma = new_sigma
+
+    return x
+
+@torch.no_grad()
+def sample_dpmpp_2m_tm(model, x, sigmas, extra_args=None, callback=None, disable=None):
+        """DPM-Solver++(2M)."""
+        extra_args = {} if extra_args is None else extra_args
+        s_in = x.new_ones([x.shape[0]])
+        sigma_fn = lambda t: t.neg().exp()
+        t_fn = lambda sigma: sigma.log().neg()
+        old_denoised = None
+
+        for i in tqdm.tqdm(self.steps, disable=disable):
+            i -= 1
+            
+            denoised = model(x, sigmas[i] * s_in, **extra_args)
+            if callback is not None:
+                callback({'x': x, 'i': i, 'sigma': sigmas[i], 'sigma_hat': sigmas[i], 'denoised': denoised})
+            t, t_next = t_fn(sigmas[i]), t_fn(sigmas[i + 1])
+            h = t_next - t
+            if old_denoised is None or sigmas[i + 1] == 0:
+                x = (sigma_fn(t_next) / sigma_fn(t)) * x - (-h).expm1() * denoised
+            else:
+                h_last = t - t_fn(sigmas[i - 1])
+                r = h_last / h
+                denoised_d = (1 + 1 / (2 * r)) * denoised - (1 / (2 * r)) * old_denoised
+                x = (sigma_fn(t_next) / sigma_fn(t)) * x - (-h).expm1() * denoised_d
+            old_denoised = denoised
+        return x