ldm/models/diffusion/dpm_solver/dpm_solver.py

import torch
import torch.nn.functional as F
import math


class NoiseScheduleVP:
    def __init__(
        self,
        schedule="discrete",
        betas=None,
        alphas_cumprod=None,
        continuous_beta_0=0.1,
        continuous_beta_1=20.0,
    ):

        if schedule not in ["discrete", "linear", "cosine"]:
            raise ValueError(
                "Unsupported noise schedule {}. The schedule needs to be 'discrete' or 'linear' or 'cosine'".format(
                    schedule
                )
            )

        self.schedule = schedule
        if schedule == "discrete":
            if betas is not None:
                log_alphas = 0.5 * torch.log(1 - betas).cumsum(dim=0)
            else:
                assert alphas_cumprod is not None
                log_alphas = 0.5 * torch.log(alphas_cumprod)
            self.total_N = len(log_alphas)
            self.T = 1.0
            self.t_array = torch.linspace(0.0, 1.0, self.total_N + 1)[1:].reshape((1, -1))
            self.log_alpha_array = log_alphas.reshape(
                (
                    1,
                    -1,
                )
            )
        else:
            self.total_N = 1000
            self.beta_0 = continuous_beta_0
            self.beta_1 = continuous_beta_1
            self.cosine_s = 0.008
            self.cosine_beta_max = 999.0
            self.cosine_t_max = (
                math.atan(self.cosine_beta_max * (1.0 + self.cosine_s) / math.pi)
                * 2.0
                * (1.0 + self.cosine_s)
                / math.pi
                - self.cosine_s
            )
            self.cosine_log_alpha_0 = math.log(math.cos(self.cosine_s / (1.0 + self.cosine_s) * math.pi / 2.0))
            self.schedule = schedule
            if schedule == "cosine":
                # For the cosine schedule, T = 1 will have numerical issues. So we manually set the ending time T.
                # Note that T = 0.9946 may be not the optimal setting. However, we find it works well.
                self.T = 0.9946
            else:
                self.T = 1.0

    def marginal_log_mean_coeff(self, t):
        """
        Compute log(alpha_t) of a given continuous-time label t in [0, T].
        """
        if self.schedule == "discrete":
            return interpolate_fn(
                t.reshape((-1, 1)), self.t_array.to(t.device), self.log_alpha_array.to(t.device)
            ).reshape((-1))
        elif self.schedule == "linear":
            return -0.25 * t**2 * (self.beta_1 - self.beta_0) - 0.5 * t * self.beta_0
        elif self.schedule == "cosine":
            log_alpha_fn = lambda s: torch.log(torch.cos((s + self.cosine_s) / (1.0 + self.cosine_s) * math.pi / 2.0))
            log_alpha_t = log_alpha_fn(t) - self.cosine_log_alpha_0
            return log_alpha_t

    def marginal_alpha(self, t):
        """
        Compute alpha_t of a given continuous-time label t in [0, T].
        """
        return torch.exp(self.marginal_log_mean_coeff(t))

    def marginal_std(self, t):
        """
        Compute sigma_t of a given continuous-time label t in [0, T].
        """
        return torch.sqrt(1.0 - torch.exp(2.0 * self.marginal_log_mean_coeff(t)))

    def marginal_lambda(self, t):
        """
        Compute lambda_t = log(alpha_t) - log(sigma_t) of a given continuous-time label t in [0, T].
        """
        log_mean_coeff = self.marginal_log_mean_coeff(t)
        log_std = 0.5 * torch.log(1.0 - torch.exp(2.0 * log_mean_coeff))
        return log_mean_coeff - log_std

    def inverse_lambda(self, lamb, return_scalar=False):
        """
        Compute the continuous-time label t in [0, T] of a given half-logSNR lambda_t.
        """
        if self.schedule == "linear":
            tmp = 2.0 * (self.beta_1 - self.beta_0) * torch.logaddexp(-2.0 * lamb, torch.zeros((1,)).to(lamb))
            Delta = self.beta_0**2 + tmp
            return tmp / (torch.sqrt(Delta) + self.beta_0) / (self.beta_1 - self.beta_0)
        elif self.schedule == "discrete":
            # check if lamb is a scalar
            if not isinstance(lamb, torch.Tensor):
                lamb = torch.tensor(lamb)
            log_alpha = -0.5 * torch.logaddexp(torch.zeros((1,)).to(lamb.device), -2.0 * lamb)
            t = interpolate_fn(
                log_alpha.reshape((-1, 1)),
                torch.flip(self.log_alpha_array.to(lamb.device), [1]),
                torch.flip(self.t_array.to(lamb.device), [1]),
            )
            if return_scalar:
                return t.reshape((-1,)).item()
            return t.reshape((-1,))
        else:
            log_alpha = -0.5 * torch.logaddexp(-2.0 * lamb, torch.zeros((1,)).to(lamb))
            t_fn = (
                lambda log_alpha_t: torch.arccos(torch.exp(log_alpha_t + self.cosine_log_alpha_0))
                * 2.0
                * (1.0 + self.cosine_s)
                / math.pi
                - self.cosine_s
            )
            t = t_fn(log_alpha)
            return t


def model_wrapper(
    model,
    noise_schedule,
    model_type="noise",
    model_kwargs={},
    guidance_type="uncond",
    condition=None,
    unconditional_condition=None,
    guidance_scale=1.0,
    classifier_fn=None,
    classifier_kwargs={},
):

    def get_model_input_time(t_continuous):
        """
        Convert the continuous-time `t_continuous` (in [epsilon, T]) to the model input time.
        For discrete-time DPMs, we convert `t_continuous` in [1 / N, 1] to `t_input` in [0, 1000 * (N - 1) / N].
        For continuous-time DPMs, we just use `t_continuous`.
        """
        if noise_schedule.schedule == "discrete":
            return (t_continuous - 1.0 / noise_schedule.total_N) * 1000.0
        else:
            return t_continuous

    def noise_pred_fn(x, t_continuous, cond=None):
        if t_continuous.reshape((-1,)).shape[0] == 1:
            t_continuous = t_continuous.expand((x.shape[0]))
        t_input = get_model_input_time(t_continuous)
        if cond is None:
            output = model(x, t_input, None, **model_kwargs)
        else:
            output = model(x, t_input, cond, **model_kwargs)
        if model_type == "noise":
            return output
        elif model_type == "x_start":
            alpha_t, sigma_t = noise_schedule.marginal_alpha(t_continuous), noise_schedule.marginal_std(t_continuous)
            dims = x.dim()
            return (x - expand_dims(alpha_t, dims) * output) / expand_dims(sigma_t, dims)
        elif model_type == "v":
            alpha_t, sigma_t = noise_schedule.marginal_alpha(t_continuous), noise_schedule.marginal_std(t_continuous)
            dims = x.dim()
            return expand_dims(alpha_t, dims) * output + expand_dims(sigma_t, dims) * x
        elif model_type == "score":
            sigma_t = noise_schedule.marginal_std(t_continuous)
            dims = x.dim()
            return -expand_dims(sigma_t, dims) * output

    def cond_grad_fn(x, t_input):
        """
        Compute the gradient of the classifier, i.e. nabla_{x} log p_t(cond | x_t).
        """
        with torch.enable_grad():
            x_in = x.detach().requires_grad_(True)
            log_prob = classifier_fn(x_in, t_input, condition, **classifier_kwargs)
            return torch.autograd.grad(log_prob.sum(), x_in)[0]

    def model_fn(x, t_continuous):
        """
        The noise predicition model function that is used for DPM-Solver.
        """
        if t_continuous.reshape((-1,)).shape[0] == 1:
            t_continuous = t_continuous.expand((x.shape[0]))
        if guidance_type == "uncond":
            return noise_pred_fn(x, t_continuous)
        elif guidance_type == "classifier":
            assert classifier_fn is not None
            t_input = get_model_input_time(t_continuous)
            cond_grad = cond_grad_fn(x, t_input)
            sigma_t = noise_schedule.marginal_std(t_continuous)
            noise = noise_pred_fn(x, t_continuous)
            return noise - guidance_scale * expand_dims(sigma_t, dims=cond_grad.dim()) * cond_grad
        elif guidance_type == "classifier-free":
            if guidance_scale == 1.0 or unconditional_condition is None:
                return noise_pred_fn(x, t_continuous, cond=condition)
            else:
                x_in = torch.cat([x] * 2)
                t_in = torch.cat([t_continuous] * 2)
                c_in = torch.cat([unconditional_condition, condition])
                noise_uncond, noise = noise_pred_fn(x_in, t_in, cond=c_in).chunk(2)
                return noise_uncond + guidance_scale * (noise - noise_uncond)

    assert model_type in ["noise", "x_start", "v"]
    assert guidance_type in ["uncond", "classifier", "classifier-free"]
    return model_fn


class DPM_Solver:
    def __init__(self, model_fn, noise_schedule, predict_x0=False, thresholding=False, max_val=1.0):
        """Construct a DPM-Solver.

        We support both the noise prediction model ("predicting epsilon") and the data prediction model ("predicting x0").
        If `predict_x0` is False, we use the solver for the noise prediction model (DPM-Solver).
        If `predict_x0` is True, we use the solver for the data prediction model (DPM-Solver++).
            In such case, we further support the "dynamic thresholding" in [1] when `thresholding` is True.
            The "dynamic thresholding" can greatly improve the sample quality for pixel-space DPMs with large guidance scales.

        Args:
            model_fn: A noise prediction model function which accepts the continuous-time input (t in [epsilon, T]):
                ``
                def model_fn(x, t_continuous):
                    return noise
                ``
            noise_schedule: A noise schedule object, such as NoiseScheduleVP.
            predict_x0: A `bool`. If true, use the data prediction model; else, use the noise prediction model.
            thresholding: A `bool`. Valid when `predict_x0` is True. Whether to use the "dynamic thresholding" in [1].
            max_val: A `float`. Valid when both `predict_x0` and `thresholding` are True. The max value for thresholding.

        [1] Chitwan Saharia, William Chan, Saurabh Saxena, Lala Li, Jay Whang, Emily Denton, Seyed Kamyar Seyed Ghasemipour, Burcu Karagol Ayan, S Sara Mahdavi, Rapha Gontijo Lopes, et al. Photorealistic text-to-image diffusion models with deep language understanding. arXiv preprint arXiv:2205.11487, 2022b.
        """
        self.model = model_fn
        self.noise_schedule = noise_schedule
        self.predict_x0 = predict_x0
        self.thresholding = thresholding
        self.max_val = max_val

    def noise_prediction_fn(self, x, t):
        """
        Return the noise prediction model.
        """
        return self.model(x, t)

    def data_prediction_fn(self, x, t):
        """
        Return the data prediction model (with thresholding).
        """
        noise = self.noise_prediction_fn(x, t)
        dims = x.dim()
        alpha_t, sigma_t = self.noise_schedule.marginal_alpha(t), self.noise_schedule.marginal_std(t)
        x0 = (x - expand_dims(sigma_t, dims) * noise) / expand_dims(alpha_t, dims)
        if self.thresholding:
            p = 0.995  # A hyperparameter in the paper of "Imagen" [1].
            s = torch.quantile(torch.abs(x0).reshape((x0.shape[0], -1)), p, dim=1)
            s = expand_dims(torch.maximum(s, self.max_val * torch.ones_like(s).to(s.device)), dims)
            x0 = torch.clamp(x0, -s, s) / s
        return x0

    def model_fn(self, x, t):
        """
        Convert the model to the noise prediction model or the data prediction model.
        """
        if self.predict_x0:
            return self.data_prediction_fn(x, t)
        else:
            return self.noise_prediction_fn(x, t)

    def get_time_steps(self, skip_type, t_T, t_0, N, device):
        """Compute the intermediate time steps for sampling.

        Args:
            skip_type: A `str`. The type for the spacing of the time steps. We support three types:
                - 'logSNR': uniform logSNR for the time steps.
                - 'time_uniform': uniform time for the time steps. (**Recommended for high-resolutional data**.)
                - 'time_quadratic': quadratic time for the time steps. (Used in DDIM for low-resolutional data.)
            t_T: A `float`. The starting time of the sampling (default is T).
            t_0: A `float`. The ending time of the sampling (default is epsilon).
            N: A `int`. The total number of the spacing of the time steps.
            device: A torch device.
        Returns:
            A pytorch tensor of the time steps, with the shape (N + 1,).
        """
        if skip_type == "logSNR":
            lambda_T = self.noise_schedule.marginal_lambda(torch.tensor(t_T).to(device))
            lambda_0 = self.noise_schedule.marginal_lambda(torch.tensor(t_0).to(device))
            logSNR_steps = torch.linspace(lambda_T.cpu().item(), lambda_0.cpu().item(), N + 1).to(device)
            return self.noise_schedule.inverse_lambda(logSNR_steps)
        elif skip_type == "time_uniform":
            return torch.linspace(t_T, t_0, N + 1).to(device)
        elif skip_type == "time_quadratic":
            t_order = 2
            t = torch.linspace(t_T ** (1.0 / t_order), t_0 ** (1.0 / t_order), N + 1).pow(t_order).to(device)
            return t
        else:
            raise ValueError(
                "Unsupported skip_type {}, need to be 'logSNR' or 'time_uniform' or 'time_quadratic'".format(skip_type)
            )

    def get_orders_and_timesteps_for_singlestep_solver(self, steps, order, skip_type, t_T, t_0, device):
        """
        Get the order of each step for sampling by the singlestep DPM-Solver.

        We combine both DPM-Solver-1,2,3 to use all the function evaluations, which is named as "DPM-Solver-fast".
        Given a fixed number of function evaluations by `steps`, the sampling procedure by DPM-Solver-fast is:
            - If order == 1:
                We take `steps` of DPM-Solver-1 (i.e. DDIM).
            - If order == 2:
                - Denote K = (steps // 2). We take K or (K + 1) intermediate time steps for sampling.
                - If steps % 2 == 0, we use K steps of DPM-Solver-2.
                - If steps % 2 == 1, we use K steps of DPM-Solver-2 and 1 step of DPM-Solver-1.
            - If order == 3:
                - Denote K = (steps // 3 + 1). We take K intermediate time steps for sampling.
                - If steps % 3 == 0, we use (K - 2) steps of DPM-Solver-3, and 1 step of DPM-Solver-2 and 1 step of DPM-Solver-1.
                - If steps % 3 == 1, we use (K - 1) steps of DPM-Solver-3 and 1 step of DPM-Solver-1.
                - If steps % 3 == 2, we use (K - 1) steps of DPM-Solver-3 and 1 step of DPM-Solver-2.

        ============================================
        Args:
            order: A `int`. The max order for the solver (2 or 3).
            steps: A `int`. The total number of function evaluations (NFE).
            skip_type: A `str`. The type for the spacing of the time steps. We support three types:
                - 'logSNR': uniform logSNR for the time steps.
                - 'time_uniform': uniform time for the time steps. (**Recommended for high-resolutional data**.)
                - 'time_quadratic': quadratic time for the time steps. (Used in DDIM for low-resolutional data.)
            t_T: A `float`. The starting time of the sampling (default is T).
            t_0: A `float`. The ending time of the sampling (default is epsilon).
            device: A torch device.
        Returns:
            orders: A list of the solver order of each step.
        """
        if order == 3:
            K = steps // 3 + 1
            if steps % 3 == 0:
                orders = [
                    3,
                ] * (
                    K - 2
                ) + [2, 1]
            elif steps % 3 == 1:
                orders = [
                    3,
                ] * (
                    K - 1
                ) + [1]
            else:
                orders = [
                    3,
                ] * (
                    K - 1
                ) + [2]
        elif order == 2:
            if steps % 2 == 0:
                K = steps // 2
                orders = [
                    2,
                ] * K
            else:
                K = steps // 2 + 1
                orders = [
                    2,
                ] * (
                    K - 1
                ) + [1]
        elif order == 1:
            K = 1
            orders = [
                1,
            ] * steps
        else:
            raise ValueError("'order' must be '1' or '2' or '3'.")
        if skip_type == "logSNR":
            # To reproduce the results in DPM-Solver paper
            timesteps_outer = self.get_time_steps(skip_type, t_T, t_0, K, device)
        else:
            timesteps_outer = self.get_time_steps(skip_type, t_T, t_0, steps, device)[
                torch.cumsum(
                    torch.tensor(
                        [
                            0,
                        ]
                        + orders
                    ),
                    dim=0,
                ).to(device)
            ]
        return timesteps_outer, orders

    def denoise_to_zero_fn(self, x, s):
        """
        Denoise at the final step, which is equivalent to solve the ODE from lambda_s to infty by first-order discretization.
        """
        return self.data_prediction_fn(x, s)

    def dpm_solver_first_update(self, x, s, t, model_s=None, return_intermediate=False):
        """
        DPM-Solver-1 (equivalent to DDIM) from time `s` to time `t`.

        Args:
            x: A pytorch tensor. The initial value at time `s`.
            s: A pytorch tensor. The starting time, with the shape (x.shape[0],).
            t: A pytorch tensor. The ending time, with the shape (x.shape[0],).
            model_s: A pytorch tensor. The model function evaluated at time `s`.
                If `model_s` is None, we evaluate the model by `x` and `s`; otherwise we directly use it.
            return_intermediate: A `bool`. If true, also return the model value at time `s`.
        Returns:
            x_t: A pytorch tensor. The approximated solution at time `t`.
        """
        ns = self.noise_schedule
        dims = x.dim()
        lambda_s, lambda_t = ns.marginal_lambda(s), ns.marginal_lambda(t)
        h = lambda_t - lambda_s
        log_alpha_s, log_alpha_t = ns.marginal_log_mean_coeff(s), ns.marginal_log_mean_coeff(t)
        sigma_s, sigma_t = ns.marginal_std(s), ns.marginal_std(t)
        alpha_t = torch.exp(log_alpha_t)

        if self.predict_x0:
            phi_1 = torch.expm1(-h)
            if model_s is None:
                model_s = self.model_fn(x, s)
            x_t = expand_dims(sigma_t / sigma_s, dims) * x - expand_dims(alpha_t * phi_1, dims) * model_s
            if return_intermediate:
                return x_t, {"model_s": model_s}
            else:
                return x_t
        else:
            phi_1 = torch.expm1(h)
            if model_s is None:
                model_s = self.model_fn(x, s)
            x_t = (
                expand_dims(torch.exp(log_alpha_t - log_alpha_s), dims) * x
                - expand_dims(sigma_t * phi_1, dims) * model_s
            )
            if return_intermediate:
                return x_t, {"model_s": model_s}
            else:
                return x_t


    def multistep_dpm_solver_second_update(self, x, model_prev_list, t_prev_list, t, solver_type="dpm_solver"):
        """
        Multistep solver DPM-Solver-2 from time `t_prev_list[-1]` to time `t`.

        Args:
            x: A pytorch tensor. The initial value at time `s`.
            model_prev_list: A list of pytorch tensor. The previous computed model values.
            t_prev_list: A list of pytorch tensor. The previous times, each time has the shape (x.shape[0],)
            t: A pytorch tensor. The ending time, with the shape (x.shape[0],).
            solver_type: either 'dpm_solver' or 'taylor'. The type for the high-order solvers.
                The type slightly impacts the performance. We recommend to use 'dpm_solver' type.
        Returns:
            x_t: A pytorch tensor. The approximated solution at time `t`.
        """
        ns = self.noise_schedule
        dims = x.dim()
        model_prev_1, model_prev_0 = model_prev_list[-2:]
        t_prev_1, t_prev_0 = t_prev_list[-2:]
        lambda_prev_1, lambda_prev_0, lambda_t = (
            ns.marginal_lambda(t_prev_1),
            ns.marginal_lambda(t_prev_0),
            ns.marginal_lambda(t),
        )
        log_alpha_prev_0, log_alpha_t = ns.marginal_log_mean_coeff(t_prev_0), ns.marginal_log_mean_coeff(t)
        sigma_prev_0, sigma_t = ns.marginal_std(t_prev_0), ns.marginal_std(t)
        alpha_t = torch.exp(log_alpha_t)

        h_0 = lambda_prev_0 - lambda_prev_1
        h = lambda_t - lambda_prev_0
        r0 = h_0 / h
        D1_0 = expand_dims(1.0 / r0, dims) * (model_prev_0 - model_prev_1)
        if self.predict_x0:
            if solver_type == "dpm_solver" or solver_type == "dpmsolver":
                x_t = (
                    expand_dims(sigma_t / sigma_prev_0, dims) * x
                    - expand_dims(alpha_t * (torch.exp(-h) - 1.0), dims) * model_prev_0
                    - 0.5 * expand_dims(alpha_t * (torch.exp(-h) - 1.0), dims) * D1_0
                )
            elif solver_type == "taylor":
                x_t = (
                    expand_dims(sigma_t / sigma_prev_0, dims) * x
                    - expand_dims(alpha_t * (torch.exp(-h) - 1.0), dims) * model_prev_0
                    + expand_dims(alpha_t * ((torch.exp(-h) - 1.0) / h + 1.0), dims) * D1_0
                )
        else:
            if solver_type == "dpm_solver" or solver_type == "dpmsolver":
                x_t = (
                    expand_dims(torch.exp(log_alpha_t - log_alpha_prev_0), dims) * x
                    - expand_dims(sigma_t * (torch.exp(h) - 1.0), dims) * model_prev_0
                    - 0.5 * expand_dims(sigma_t * (torch.exp(h) - 1.0), dims) * D1_0
                )
            elif solver_type == "taylor":
                x_t = (
                    expand_dims(torch.exp(log_alpha_t - log_alpha_prev_0), dims) * x
                    - expand_dims(sigma_t * (torch.exp(h) - 1.0), dims) * model_prev_0
                    - expand_dims(sigma_t * ((torch.exp(h) - 1.0) / h - 1.0), dims) * D1_0
                )
        return x_t


    def multistep_dpm_solver_third_update(self, x, model_prev_list, t_prev_list, t, solver_type="dpm_solver"):
        """
        Multistep solver DPM-Solver-3 from time `t_prev_list[-1]` to time `t`.

        Args:
            x: A pytorch tensor. The initial value at time `s`.
            model_prev_list: A list of pytorch tensor. The previous computed model values.
            t_prev_list: A list of pytorch tensor. The previous times, each time has the shape (x.shape[0],)
            t: A pytorch tensor. The ending time, with the shape (x.shape[0],).
            solver_type: either 'dpm_solver' or 'taylor'. The type for the high-order solvers.
                The type slightly impacts the performance. We recommend to use 'dpm_solver' type.
        Returns:
            x_t: A pytorch tensor. The approximated solution at time `t`.
        """
        ns = self.noise_schedule
        dims = x.dim()
        model_prev_2, model_prev_1, model_prev_0 = model_prev_list
        t_prev_2, t_prev_1, t_prev_0 = t_prev_list
        lambda_prev_2, lambda_prev_1, lambda_prev_0, lambda_t = (
            ns.marginal_lambda(t_prev_2),
            ns.marginal_lambda(t_prev_1),
            ns.marginal_lambda(t_prev_0),
            ns.marginal_lambda(t),
        )
        log_alpha_prev_0, log_alpha_t = ns.marginal_log_mean_coeff(t_prev_0), ns.marginal_log_mean_coeff(t)
        sigma_prev_0, sigma_t = ns.marginal_std(t_prev_0), ns.marginal_std(t)
        alpha_t = torch.exp(log_alpha_t)

        h_1 = lambda_prev_1 - lambda_prev_2
        h_0 = lambda_prev_0 - lambda_prev_1
        h = lambda_t - lambda_prev_0
        r0, r1 = h_0 / h, h_1 / h
        D1_0 = expand_dims(1.0 / r0, dims) * (model_prev_0 - model_prev_1)
        D1_1 = expand_dims(1.0 / r1, dims) * (model_prev_1 - model_prev_2)
        D1 = D1_0 + expand_dims(r0 / (r0 + r1), dims) * (D1_0 - D1_1)
        D2 = expand_dims(1.0 / (r0 + r1), dims) * (D1_0 - D1_1)
        if self.predict_x0:
            x_t = (
                expand_dims(sigma_t / sigma_prev_0, dims) * x
                - expand_dims(alpha_t * (torch.exp(-h) - 1.0), dims) * model_prev_0
                + expand_dims(alpha_t * ((torch.exp(-h) - 1.0) / h + 1.0), dims) * D1
                - expand_dims(alpha_t * ((torch.exp(-h) - 1.0 + h) / h**2 - 0.5), dims) * D2
            )
        else:
            x_t = (
                expand_dims(torch.exp(log_alpha_t - log_alpha_prev_0), dims) * x
                - expand_dims(sigma_t * (torch.exp(h) - 1.0), dims) * model_prev_0
                - expand_dims(sigma_t * ((torch.exp(h) - 1.0) / h - 1.0), dims) * D1
                - expand_dims(sigma_t * ((torch.exp(h) - 1.0 - h) / h**2 - 0.5), dims) * D2
            )
        return x_t

    def multistep_dpm_solver_update(self, x, model_prev_list, t_prev_list, t, order, solver_type="dpm_solver"):
        """
        Multistep DPM-Solver with the order `order` from time `t_prev_list[-1]` to time `t`.

        Args:
            x: A pytorch tensor. The initial value at time `s`.
            model_prev_list: A list of pytorch tensor. The previous computed model values.
            t_prev_list: A list of pytorch tensor. The previous times, each time has the shape (x.shape[0],)
            t: A pytorch tensor. The ending time, with the shape (x.shape[0],).
            order: A `int`. The order of DPM-Solver. We only support order == 1 or 2 or 3.
            solver_type: either 'dpm_solver' or 'taylor'. The type for the high-order solvers.
                The type slightly impacts the performance. We recommend to use 'dpm_solver' type.
        Returns:
            x_t: A pytorch tensor. The approximated solution at time `t`.
        """
        if order == 1:
            return self.dpm_solver_first_update(x, t_prev_list[-1], t, model_s=model_prev_list[-1])
        elif order == 2:
            return self.multistep_dpm_solver_second_update(x, model_prev_list, t_prev_list, t, solver_type=solver_type)
        elif order == 3:
            return self.multistep_dpm_solver_third_update(x, model_prev_list, t_prev_list, t, solver_type=solver_type)
        else:
            raise ValueError("Solver order must be 1 or 2 or 3, got {}".format(order))

    def dpm_solver_adaptive(
        self, x, order, t_T, t_0, h_init=0.05, atol=0.0078, rtol=0.05, theta=0.9, t_err=1e-5, solver_type="dpm_solver"
    ):
        """
        The adaptive step size solver based on singlestep DPM-Solver.

        Args:
            x: A pytorch tensor. The initial value at time `t_T`.
            order: A `int`. The (higher) order of the solver. We only support order == 2 or 3.
            t_T: A `float`. The starting time of the sampling (default is T).
            t_0: A `float`. The ending time of the sampling (default is epsilon).
            h_init: A `float`. The initial step size (for logSNR).
            atol: A `float`. The absolute tolerance of the solver. For image data, the default setting is 0.0078, followed [1].
            rtol: A `float`. The relative tolerance of the solver. The default setting is 0.05.
            theta: A `float`. The safety hyperparameter for adapting the step size. The default setting is 0.9, followed [1].
            t_err: A `float`. The tolerance for the time. We solve the diffusion ODE until the absolute error between the
                current time and `t_0` is less than `t_err`. The default setting is 1e-5.
            solver_type: either 'dpm_solver' or 'taylor'. The type for the high-order solvers.
                The type slightly impacts the performance. We recommend to use 'dpm_solver' type.
        Returns:
            x_0: A pytorch tensor. The approximated solution at time `t_0`.

        [1] A. Jolicoeur-Martineau, K. Li, R. Piché-Taillefer, T. Kachman, and I. Mitliagkas, "Gotta go fast when generating data with score-based models," arXiv preprint arXiv:2105.14080, 2021.
        """
        ns = self.noise_schedule
        s = t_T * torch.ones((x.shape[0],)).to(x)
        lambda_s = ns.marginal_lambda(s)
        lambda_0 = ns.marginal_lambda(t_0 * torch.ones_like(s).to(x))
        h = h_init * torch.ones_like(s).to(x)
        x_prev = x
        nfe = 0
        if order == 2:
            r1 = 0.5
            lower_update = lambda x, s, t: self.dpm_solver_first_update(x, s, t, return_intermediate=True)
            higher_update = lambda x, s, t, **kwargs: self.singlestep_dpm_solver_second_update(
                x, s, t, r1=r1, solver_type=solver_type, **kwargs
            )
        elif order == 3:
            r1, r2 = 1.0 / 3.0, 2.0 / 3.0
            lower_update = lambda x, s, t: self.singlestep_dpm_solver_second_update(
                x, s, t, r1=r1, return_intermediate=True, solver_type=solver_type
            )
            higher_update = lambda x, s, t, **kwargs: self.singlestep_dpm_solver_third_update(
                x, s, t, r1=r1, r2=r2, solver_type=solver_type, **kwargs
            )
        else:
            raise ValueError("For adaptive step size solver, order must be 2 or 3, got {}".format(order))
        while torch.abs((s - t_0)).mean() > t_err:
            t = ns.inverse_lambda(lambda_s + h)
            x_lower, lower_noise_kwargs = lower_update(x, s, t)
            x_higher = higher_update(x, s, t, **lower_noise_kwargs)
            delta = torch.max(torch.ones_like(x).to(x) * atol, rtol * torch.max(torch.abs(x_lower), torch.abs(x_prev)))
            norm_fn = lambda v: torch.sqrt(torch.square(v.reshape((v.shape[0], -1))).mean(dim=-1, keepdim=True))
            E = norm_fn((x_higher - x_lower) / delta).max()
            if torch.all(E <= 1.0):
                x = x_higher
                s = t
                x_prev = x_lower
                lambda_s = ns.marginal_lambda(s)
            h = torch.min(theta * h * torch.float_power(E, -1.0 / order).float(), lambda_0 - lambda_s)
            nfe += order
        print("adaptive solver nfe", nfe)
        return x

    def sample(
        self,
        x,
        steps=20,
        t_start=None,
        t_end=None,
        order=3,
        skip_type="time_uniform",
        method="singlestep",
        lower_order_final=True,
        denoise_to_zero=False,
        solver_type="dpm_solver",
        atol=0.0078,
        rtol=0.05,
        flags=None,
    ):
        device = x.device  
        with torch.no_grad():
            if flags.learn:
                load_from = f"{flags.log_path}/NFE-{steps}-256LSUN-dpmsolver++-{order}-decode/best.pt"
                timesteps = torch.load(load_from)['best_t_steps'].to(x.device)
                if flags:
                    length = timesteps.shape[0] // 2
                    timesteps2 = timesteps[length:]
                    timesteps = timesteps[:length]
                else:
                    timesteps2 = timesteps   
            else: 
                t_0 = 1.0 / self.noise_schedule.total_N if t_end is None else t_end
                t_T = self.noise_schedule.T if t_start is None else t_start
                timesteps = self.get_time_steps(skip_type=skip_type, t_T=t_T, t_0=t_0, N=steps, device=device)
                timesteps2 = timesteps
                assert timesteps.shape[0] - 1 == steps
            
            def one_step(t1, t2, t_prev_list, model_prev_list, step, x_next, order, first=True):
                x_next = self.multistep_dpm_solver_update(x_next, model_prev_list, t_prev_list, t1, step, solver_type="dpmsolver")
                model_x_next = self.model_fn(x_next, t2)
                update_lists(t_prev_list, model_prev_list, t1, model_x_next, order, first=first)
                return x_next
            
            def update_lists(t_list, model_list, t_, model_x, order, first=False):
                if first:
                    t_list.append(t_)
                    model_list.append(model_x)
                    return
                for m in range(order - 1):
                    t_list[m] = t_list[m + 1]
                    model_list[m] = model_list[m + 1]
                t_list[-1] = t_
                model_list[-1] = model_x
                
            timesteps1 = timesteps
            step = 0
            vec_t1 = timesteps1[0].expand((x.shape[0]))
            vec_t2 = timesteps2[0].expand((x.shape[0]))
            t_prev_list = [vec_t1]
            model_prev_list = [self.model_fn(x, vec_t2)]
                
            
            for step in range(1, order):
                vec_t1 = timesteps1[step].expand(x.shape[0])
                vec_t2 = timesteps2[step].expand(x.shape[0])
                x = one_step(vec_t1, vec_t2, t_prev_list, model_prev_list, step, x, order, first=True)
            
            for step in range(order, steps + 1):
                step_order = min(order, steps + 1 - step)
                vec_t1 = timesteps1[step].expand(x.shape[0])
                vec_t2 = timesteps2[step].expand(x.shape[0])
                x = one_step(vec_t1, vec_t2, t_prev_list, model_prev_list, step_order, x, order, first=False)

        return x


#############################################################
# other utility functions
#############################################################


def interpolate_fn(x, xp, yp):
    """
    A piecewise linear function y = f(x), using xp and yp as keypoints.
    We implement f(x) in a differentiable way (i.e. applicable for autograd).
    The function f(x) is well-defined for all x-axis. (For x beyond the bounds of xp, we use the outmost points of xp to define the linear function.)

    Args:
        x: PyTorch tensor with shape [N, C], where N is the batch size, C is the number of channels (we use C = 1 for DPM-Solver).
        xp: PyTorch tensor with shape [C, K], where K is the number of keypoints.
        yp: PyTorch tensor with shape [C, K].
    Returns:
        The function values f(x), with shape [N, C].
    """
    N, K = x.shape[0], xp.shape[1]
    all_x = torch.cat([x.unsqueeze(2), xp.unsqueeze(0).repeat((N, 1, 1))], dim=2)
    sorted_all_x, x_indices = torch.sort(all_x, dim=2)
    x_idx = torch.argmin(x_indices, dim=2)
    cand_start_idx = x_idx - 1
    start_idx = torch.where(
        torch.eq(x_idx, 0),
        torch.tensor(1, device=x.device),
        torch.where(
            torch.eq(x_idx, K),
            torch.tensor(K - 2, device=x.device),
            cand_start_idx,
        ),
    )
    end_idx = torch.where(torch.eq(start_idx, cand_start_idx), start_idx + 2, start_idx + 1)
    start_x = torch.gather(sorted_all_x, dim=2, index=start_idx.unsqueeze(2)).squeeze(2)
    end_x = torch.gather(sorted_all_x, dim=2, index=end_idx.unsqueeze(2)).squeeze(2)
    start_idx2 = torch.where(
        torch.eq(x_idx, 0),
        torch.tensor(0, device=x.device),
        torch.where(
            torch.eq(x_idx, K),
            torch.tensor(K - 2, device=x.device),
            cand_start_idx,
        ),
    )
    y_positions_expanded = yp.unsqueeze(0).expand(N, -1, -1)
    start_y = torch.gather(y_positions_expanded, dim=2, index=start_idx2.unsqueeze(2)).squeeze(2)
    end_y = torch.gather(y_positions_expanded, dim=2, index=(start_idx2 + 1).unsqueeze(2)).squeeze(2)
    cand = start_y + (x - start_x) * (end_y - start_y) / (end_x - start_x)
    return cand


def expand_dims(v, dims):
    """
    Expand the tensor `v` to the dim `dims`.

    Args:
        `v`: a PyTorch tensor with shape [N].
        `dim`: a `int`.
    Returns:
        a PyTorch tensor with shape [N, 1, 1, ..., 1] and the total dimension is `dims`.
    """
    return v[(...,) + (None,) * (dims - 1)]