stable-diffusion-webui/modules/models/diffusion/uni_pc/uni_pc.py

import torch
import math
import tqdm


class NoiseScheduleVP:
    def __init__(
            self,
            schedule='discrete',
            betas=None,
            alphas_cumprod=None,
            continuous_beta_0=0.1,
            continuous_beta_1=20.,
        ):
        """Create a wrapper class for the forward SDE (VP type).

        ***
        Update: We support discrete-time diffusion models by implementing a picewise linear interpolation for log_alpha_t.
                We recommend to use schedule='discrete' for the discrete-time diffusion models, especially for high-resolution images.
        ***

        The forward SDE ensures that the condition distribution q_{t|0}(x_t | x_0) = N ( alpha_t * x_0, sigma_t^2 * I ).
        We further define lambda_t = log(alpha_t) - log(sigma_t), which is the half-logSNR (described in the DPM-Solver paper).
        Therefore, we implement the functions for computing alpha_t, sigma_t and lambda_t. For t in [0, T], we have:

            log_alpha_t = self.marginal_log_mean_coeff(t)
            sigma_t = self.marginal_std(t)
            lambda_t = self.marginal_lambda(t)

        Moreover, as lambda(t) is an invertible function, we also support its inverse function:

            t = self.inverse_lambda(lambda_t)

        ===============================================================

        We support both discrete-time DPMs (trained on n = 0, 1, ..., N-1) and continuous-time DPMs (trained on t in [t_0, T]).

        1. For discrete-time DPMs:

            For discrete-time DPMs trained on n = 0, 1, ..., N-1, we convert the discrete steps to continuous time steps by:
                t_i = (i + 1) / N
            e.g. for N = 1000, we have t_0 = 1e-3 and T = t_{N-1} = 1.
            We solve the corresponding diffusion ODE from time T = 1 to time t_0 = 1e-3.

            Args:
                betas: A `torch.Tensor`. The beta array for the discrete-time DPM. (See the original DDPM paper for details)
                alphas_cumprod: A `torch.Tensor`. The cumprod alphas for the discrete-time DPM. (See the original DDPM paper for details)

            Note that we always have alphas_cumprod = cumprod(betas). Therefore, we only need to set one of `betas` and `alphas_cumprod`.

            **Important**:  Please pay special attention for the args for `alphas_cumprod`:
                The `alphas_cumprod` is the \hat{alpha_n} arrays in the notations of DDPM. Specifically, DDPMs assume that
                    q_{t_n | 0}(x_{t_n} | x_0) = N ( \sqrt{\hat{alpha_n}} * x_0, (1 - \hat{alpha_n}) * I ).
                Therefore, the notation \hat{alpha_n} is different from the notation alpha_t in DPM-Solver. In fact, we have
                    alpha_{t_n} = \sqrt{\hat{alpha_n}},
                and
                    log(alpha_{t_n}) = 0.5 * log(\hat{alpha_n}).


        2. For continuous-time DPMs:

            We support two types of VPSDEs: linear (DDPM) and cosine (improved-DDPM). The hyperparameters for the noise
            schedule are the default settings in DDPM and improved-DDPM:

            Args:
                beta_min: A `float` number. The smallest beta for the linear schedule.
                beta_max: A `float` number. The largest beta for the linear schedule.
                cosine_s: A `float` number. The hyperparameter in the cosine schedule.
                cosine_beta_max: A `float` number. The hyperparameter in the cosine schedule.
                T: A `float` number. The ending time of the forward process.

        ===============================================================

        Args:
            schedule: A `str`. The noise schedule of the forward SDE. 'discrete' for discrete-time DPMs,
                    'linear' or 'cosine' for continuous-time DPMs.
        Returns:
            A wrapper object of the forward SDE (VP type).

        ===============================================================

        Example:

        # For discrete-time DPMs, given betas (the beta array for n = 0, 1, ..., N - 1):
        >>> ns = NoiseScheduleVP('discrete', betas=betas)

        # For discrete-time DPMs, given alphas_cumprod (the \hat{alpha_n} array for n = 0, 1, ..., N - 1):
        >>> ns = NoiseScheduleVP('discrete', alphas_cumprod=alphas_cumprod)

        # For continuous-time DPMs (VPSDE), linear schedule:
        >>> ns = NoiseScheduleVP('linear', continuous_beta_0=0.1, continuous_beta_1=20.)

        """

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

        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.
            self.t_array = torch.linspace(0., 1., 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.
            self.cosine_t_max = math.atan(self.cosine_beta_max * (1. + self.cosine_s) / math.pi) * 2. * (1. + self.cosine_s) / math.pi - self.cosine_s
            self.cosine_log_alpha_0 = math.log(math.cos(self.cosine_s / (1. + self.cosine_s) * math.pi / 2.))
            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.

    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. + self.cosine_s) * math.pi / 2.))
            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. - torch.exp(2. * 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. - torch.exp(2. * log_mean_coeff))
        return log_mean_coeff - log_std

    def inverse_lambda(self, lamb):
        """
        Compute the continuous-time label t in [0, T] of a given half-logSNR lambda_t.
        """
        if self.schedule == 'linear':
            tmp = 2. * (self.beta_1 - self.beta_0) * torch.logaddexp(-2. * 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':
            log_alpha = -0.5 * torch.logaddexp(torch.zeros((1,)).to(lamb.device), -2. * 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]))
            return t.reshape((-1,))
        else:
            log_alpha = -0.5 * torch.logaddexp(-2. * 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. * (1. + 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=None,
    guidance_type="uncond",
    #condition=None,
    #unconditional_condition=None,
    guidance_scale=1.,
    classifier_fn=None,
    classifier_kwargs=None,
):
    """Create a wrapper function for the noise prediction model.

    DPM-Solver needs to solve the continuous-time diffusion ODEs. For DPMs trained on discrete-time labels, we need to
    firstly wrap the model function to a noise prediction model that accepts the continuous time as the input.

    We support four types of the diffusion model by setting `model_type`:

        1. "noise": noise prediction model. (Trained by predicting noise).

        2. "x_start": data prediction model. (Trained by predicting the data x_0 at time 0).

        3. "v": velocity prediction model. (Trained by predicting the velocity).
            The "v" prediction is derivation detailed in Appendix D of [1], and is used in Imagen-Video [2].

            [1] Salimans, Tim, and Jonathan Ho. "Progressive distillation for fast sampling of diffusion models."
                arXiv preprint arXiv:2202.00512 (2022).
            [2] Ho, Jonathan, et al. "Imagen Video: High Definition Video Generation with Diffusion Models."
                arXiv preprint arXiv:2210.02303 (2022).

        4. "score": marginal score function. (Trained by denoising score matching).
            Note that the score function and the noise prediction model follows a simple relationship:
            ```
                noise(x_t, t) = -sigma_t * score(x_t, t)
            ```

    We support three types of guided sampling by DPMs by setting `guidance_type`:
        1. "uncond": unconditional sampling by DPMs.
            The input `model` has the following format:
            ``
                model(x, t_input, **model_kwargs) -> noise | x_start | v | score
            ``

        2. "classifier": classifier guidance sampling [3] by DPMs and another classifier.
            The input `model` has the following format:
            ``
                model(x, t_input, **model_kwargs) -> noise | x_start | v | score
            ``

            The input `classifier_fn` has the following format:
            ``
                classifier_fn(x, t_input, cond, **classifier_kwargs) -> logits(x, t_input, cond)
            ``

            [3] P. Dhariwal and A. Q. Nichol, "Diffusion models beat GANs on image synthesis,"
                in Advances in Neural Information Processing Systems, vol. 34, 2021, pp. 8780-8794.

        3. "classifier-free": classifier-free guidance sampling by conditional DPMs.
            The input `model` has the following format:
            ``
                model(x, t_input, cond, **model_kwargs) -> noise | x_start | v | score
            ``
            And if cond == `unconditional_condition`, the model output is the unconditional DPM output.

            [4] Ho, Jonathan, and Tim Salimans. "Classifier-free diffusion guidance."
                arXiv preprint arXiv:2207.12598 (2022).


    The `t_input` is the time label of the model, which may be discrete-time labels (i.e. 0 to 999)
    or continuous-time labels (i.e. epsilon to T).

    We wrap the model function to accept only `x` and `t_continuous` as inputs, and outputs the predicted noise:
    ``
        def model_fn(x, t_continuous) -> noise:
            t_input = get_model_input_time(t_continuous)
            return noise_pred(model, x, t_input, **model_kwargs)
    ``
    where `t_continuous` is the continuous time labels (i.e. epsilon to T). And we use `model_fn` for DPM-Solver.

    ===============================================================

    Args:
        model: A diffusion model with the corresponding format described above.
        noise_schedule: A noise schedule object, such as NoiseScheduleVP.
        model_type: A `str`. The parameterization type of the diffusion model.
                    "noise" or "x_start" or "v" or "score".
        model_kwargs: A `dict`. A dict for the other inputs of the model function.
        guidance_type: A `str`. The type of the guidance for sampling.
                    "uncond" or "classifier" or "classifier-free".
        condition: A pytorch tensor. The condition for the guided sampling.
                    Only used for "classifier" or "classifier-free" guidance type.
        unconditional_condition: A pytorch tensor. The condition for the unconditional sampling.
                    Only used for "classifier-free" guidance type.
        guidance_scale: A `float`. The scale for the guided sampling.
        classifier_fn: A classifier function. Only used for the classifier guidance.
        classifier_kwargs: A `dict`. A dict for the other inputs of the classifier function.
    Returns:
        A noise prediction model that accepts the noised data and the continuous time as the inputs.
    """

    model_kwargs = model_kwargs or {}
    classifier_kwargs = classifier_kwargs or {}

    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. / noise_schedule.total_N) * 1000.
        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, condition):
        """
        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, condition, unconditional_condition):
        """
        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, condition)
            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. 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)
                if isinstance(condition, dict):
                    assert isinstance(unconditional_condition, dict)
                    c_in = {}
                    for k in condition:
                        if isinstance(condition[k], list):
                            c_in[k] = [torch.cat([
                                unconditional_condition[k][i],
                                condition[k][i]]) for i in range(len(condition[k]))]
                        else:
                            c_in[k] = torch.cat([
                                unconditional_condition[k],
                                condition[k]])
                elif isinstance(condition, list):
                    c_in = []
                    assert isinstance(unconditional_condition, list)
                    for i in range(len(condition)):
                        c_in.append(torch.cat([unconditional_condition[i], condition[i]]))
                else:
                    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 UniPC:
    def __init__(
        self,
        model_fn,
        noise_schedule,
        predict_x0=True,
        thresholding=False,
        max_val=1.,
        variant='bh1',
        condition=None,
        unconditional_condition=None,
        before_sample=None,
        after_sample=None,
        after_update=None
    ):
        """Construct a UniPC.

        We support both data_prediction and noise_prediction.
        """
        self.model_fn_ = model_fn
        self.noise_schedule = noise_schedule
        self.variant = variant
        self.predict_x0 = predict_x0
        self.thresholding = thresholding
        self.max_val = max_val
        self.condition = condition
        self.unconditional_condition = unconditional_condition
        self.before_sample = before_sample
        self.after_sample = after_sample
        self.after_update = after_update

    def dynamic_thresholding_fn(self, x0, t=None):
        """
        The dynamic thresholding method.
        """
        dims = x0.dim()
        p = self.dynamic_thresholding_ratio
        s = torch.quantile(torch.abs(x0).reshape((x0.shape[0], -1)), p, dim=1)
        s = expand_dims(torch.maximum(s, self.thresholding_max_val * torch.ones_like(s).to(s.device)), dims)
        x0 = torch.clamp(x0, -s, s) / s
        return x0

    def model(self, x, t):
        cond = self.condition
        uncond = self.unconditional_condition
        if self.before_sample is not None:
            x, t, cond, uncond = self.before_sample(x, t, cond, uncond)
        res = self.model_fn_(x, t, cond, uncond)
        if self.after_sample is not None:
            x, t, cond, uncond, res = self.after_sample(x, t, cond, uncond, res)

        if isinstance(res, tuple):
            # (None, pred_x0)
            res = res[1]

        return res

    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.
        """
        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. / t_order), t_0**(1. / t_order), N + 1).pow(t_order).to(device)
            return t
        else:
            raise ValueError(f"Unsupported skip_type {skip_type}, need to be 'logSNR' or 'time_uniform' or 'time_quadratic'")

    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.
        """
        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 = steps
            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), 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 multistep_uni_pc_update(self, x, model_prev_list, t_prev_list, t, order, **kwargs):
        if len(t.shape) == 0:
            t = t.view(-1)
        if 'bh' in self.variant:
            return self.multistep_uni_pc_bh_update(x, model_prev_list, t_prev_list, t, order, **kwargs)
        else:
            assert self.variant == 'vary_coeff'
            return self.multistep_uni_pc_vary_update(x, model_prev_list, t_prev_list, t, order, **kwargs)

    def multistep_uni_pc_vary_update(self, x, model_prev_list, t_prev_list, t, order, use_corrector=True):
        #print(f'using unified predictor-corrector with order {order} (solver type: vary coeff)')
        ns = self.noise_schedule
        assert order <= len(model_prev_list)

        # first compute rks
        t_prev_0 = t_prev_list[-1]
        lambda_prev_0 = ns.marginal_lambda(t_prev_0)
        lambda_t = ns.marginal_lambda(t)
        model_prev_0 = model_prev_list[-1]
        sigma_prev_0, sigma_t = ns.marginal_std(t_prev_0), ns.marginal_std(t)
        log_alpha_t = ns.marginal_log_mean_coeff(t)
        alpha_t = torch.exp(log_alpha_t)

        h = lambda_t - lambda_prev_0

        rks = []
        D1s = []
        for i in range(1, order):
            t_prev_i = t_prev_list[-(i + 1)]
            model_prev_i = model_prev_list[-(i + 1)]
            lambda_prev_i = ns.marginal_lambda(t_prev_i)
            rk = (lambda_prev_i - lambda_prev_0) / h
            rks.append(rk)
            D1s.append((model_prev_i - model_prev_0) / rk)

        rks.append(1.)
        rks = torch.tensor(rks, device=x.device)

        K = len(rks)
        # build C matrix
        C = []

        col = torch.ones_like(rks)
        for k in range(1, K + 1):
            C.append(col)
            col = col * rks / (k + 1)
        C = torch.stack(C, dim=1)

        if len(D1s) > 0:
            D1s = torch.stack(D1s, dim=1) # (B, K)
            C_inv_p = torch.linalg.inv(C[:-1, :-1])
            A_p = C_inv_p

        if use_corrector:
            #print('using corrector')
            C_inv = torch.linalg.inv(C)
            A_c = C_inv

        hh = -h if self.predict_x0 else h
        h_phi_1 = torch.expm1(hh)
        h_phi_ks = []
        factorial_k = 1
        h_phi_k = h_phi_1
        for k in range(1, K + 2):
            h_phi_ks.append(h_phi_k)
            h_phi_k = h_phi_k / hh - 1 / factorial_k
            factorial_k *= (k + 1)

        model_t = None
        if self.predict_x0:
            x_t_ = (
                sigma_t / sigma_prev_0 * x
                - alpha_t * h_phi_1 * model_prev_0
            )
            # now predictor
            x_t = x_t_
            if len(D1s) > 0:
                # compute the residuals for predictor
                for k in range(K - 1):
                    x_t = x_t - alpha_t * h_phi_ks[k + 1] * torch.einsum('bkchw,k->bchw', D1s, A_p[k])
            # now corrector
            if use_corrector:
                model_t = self.model_fn(x_t, t)
                D1_t = (model_t - model_prev_0)
                x_t = x_t_
                k = 0
                for k in range(K - 1):
                    x_t = x_t - alpha_t * h_phi_ks[k + 1] * torch.einsum('bkchw,k->bchw', D1s, A_c[k][:-1])
                x_t = x_t - alpha_t * h_phi_ks[K] * (D1_t * A_c[k][-1])
        else:
            log_alpha_prev_0, log_alpha_t = ns.marginal_log_mean_coeff(t_prev_0), ns.marginal_log_mean_coeff(t)
            x_t_ = (
                (torch.exp(log_alpha_t - log_alpha_prev_0)) * x
                - (sigma_t * h_phi_1) * model_prev_0
            )
            # now predictor
            x_t = x_t_
            if len(D1s) > 0:
                # compute the residuals for predictor
                for k in range(K - 1):
                    x_t = x_t - sigma_t * h_phi_ks[k + 1] * torch.einsum('bkchw,k->bchw', D1s, A_p[k])
            # now corrector
            if use_corrector:
                model_t = self.model_fn(x_t, t)
                D1_t = (model_t - model_prev_0)
                x_t = x_t_
                k = 0
                for k in range(K - 1):
                    x_t = x_t - sigma_t * h_phi_ks[k + 1] * torch.einsum('bkchw,k->bchw', D1s, A_c[k][:-1])
                x_t = x_t - sigma_t * h_phi_ks[K] * (D1_t * A_c[k][-1])
        return x_t, model_t

    def multistep_uni_pc_bh_update(self, x, model_prev_list, t_prev_list, t, order, x_t=None, use_corrector=True):
        #print(f'using unified predictor-corrector with order {order} (solver type: B(h))')
        ns = self.noise_schedule
        assert order <= len(model_prev_list)
        dims = x.dim()

        # first compute rks
        t_prev_0 = t_prev_list[-1]
        lambda_prev_0 = ns.marginal_lambda(t_prev_0)
        lambda_t = ns.marginal_lambda(t)
        model_prev_0 = model_prev_list[-1]
        sigma_prev_0, sigma_t = ns.marginal_std(t_prev_0), ns.marginal_std(t)
        log_alpha_prev_0, log_alpha_t = ns.marginal_log_mean_coeff(t_prev_0), ns.marginal_log_mean_coeff(t)
        alpha_t = torch.exp(log_alpha_t)

        h = lambda_t - lambda_prev_0

        rks = []
        D1s = []
        for i in range(1, order):
            t_prev_i = t_prev_list[-(i + 1)]
            model_prev_i = model_prev_list[-(i + 1)]
            lambda_prev_i = ns.marginal_lambda(t_prev_i)
            rk = ((lambda_prev_i - lambda_prev_0) / h)[0]
            rks.append(rk)
            D1s.append((model_prev_i - model_prev_0) / rk)

        rks.append(1.)
        rks = torch.tensor(rks, device=x.device)

        R = []
        b = []

        hh = -h[0] if self.predict_x0 else h[0]
        h_phi_1 = torch.expm1(hh) # h\phi_1(h) = e^h - 1
        h_phi_k = h_phi_1 / hh - 1

        factorial_i = 1

        if self.variant == 'bh1':
            B_h = hh
        elif self.variant == 'bh2':
            B_h = torch.expm1(hh)
        else:
            raise NotImplementedError()

        for i in range(1, order + 1):
            R.append(torch.pow(rks, i - 1))
            b.append(h_phi_k * factorial_i / B_h)
            factorial_i *= (i + 1)
            h_phi_k = h_phi_k / hh - 1 / factorial_i

        R = torch.stack(R)
        b = torch.tensor(b, device=x.device)

        # now predictor
        use_predictor = len(D1s) > 0 and x_t is None
        if len(D1s) > 0:
            D1s = torch.stack(D1s, dim=1) # (B, K)
            if x_t is None:
                # for order 2, we use a simplified version
                if order == 2:
                    rhos_p = torch.tensor([0.5], device=b.device)
                else:
                    rhos_p = torch.linalg.solve(R[:-1, :-1], b[:-1])
        else:
            D1s = None

        if use_corrector:
            #print('using corrector')
            # for order 1, we use a simplified version
            if order == 1:
                rhos_c = torch.tensor([0.5], device=b.device)
            else:
                rhos_c = torch.linalg.solve(R, b)

        model_t = None
        if self.predict_x0:
            x_t_ = (
                expand_dims(sigma_t / sigma_prev_0, dims) * x
                - expand_dims(alpha_t * h_phi_1, dims)* model_prev_0
            )

            if x_t is None:
                if use_predictor:
                    pred_res = torch.einsum('k,bkchw->bchw', rhos_p, D1s)
                else:
                    pred_res = 0
                x_t = x_t_ - expand_dims(alpha_t * B_h, dims) * pred_res

            if use_corrector:
                model_t = self.model_fn(x_t, t)
                if D1s is not None:
                    corr_res = torch.einsum('k,bkchw->bchw', rhos_c[:-1], D1s)
                else:
                    corr_res = 0
                D1_t = (model_t - model_prev_0)
                x_t = x_t_ - expand_dims(alpha_t * B_h, dims) * (corr_res + rhos_c[-1] * D1_t)
        else:
            x_t_ = (
                expand_dims(torch.exp(log_alpha_t - log_alpha_prev_0), dims) * x
                - expand_dims(sigma_t * h_phi_1, dims) * model_prev_0
            )
            if x_t is None:
                if use_predictor:
                    pred_res = torch.einsum('k,bkchw->bchw', rhos_p, D1s)
                else:
                    pred_res = 0
                x_t = x_t_ - expand_dims(sigma_t * B_h, dims) * pred_res

            if use_corrector:
                model_t = self.model_fn(x_t, t)
                if D1s is not None:
                    corr_res = torch.einsum('k,bkchw->bchw', rhos_c[:-1], D1s)
                else:
                    corr_res = 0
                D1_t = (model_t - model_prev_0)
                x_t = x_t_ - expand_dims(sigma_t * B_h, dims) * (corr_res + rhos_c[-1] * D1_t)
        return x_t, model_t


    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, corrector=False,
    ):
        t_0 = 1. / 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
        device = x.device
        if method == 'multistep':
            assert steps >= order, "UniPC order must be < sampling steps"
            timesteps = self.get_time_steps(skip_type=skip_type, t_T=t_T, t_0=t_0, N=steps, device=device)
            #print(f"Running UniPC Sampling with {timesteps.shape[0]} timesteps, order {order}")
            assert timesteps.shape[0] - 1 == steps
            with torch.no_grad():
                vec_t = timesteps[0].expand((x.shape[0]))
                model_prev_list = [self.model_fn(x, vec_t)]
                t_prev_list = [vec_t]
                with tqdm.tqdm(total=steps) as pbar:
                    # Init the first `order` values by lower order multistep DPM-Solver.
                    for init_order in range(1, order):
                        vec_t = timesteps[init_order].expand(x.shape[0])
                        x, model_x = self.multistep_uni_pc_update(x, model_prev_list, t_prev_list, vec_t, init_order, use_corrector=True)
                        if model_x is None:
                            model_x = self.model_fn(x, vec_t)
                        if self.after_update is not None:
                            self.after_update(x, model_x)
                        model_prev_list.append(model_x)
                        t_prev_list.append(vec_t)
                        pbar.update()

                    for step in range(order, steps + 1):
                        vec_t = timesteps[step].expand(x.shape[0])
                        if lower_order_final:
                            step_order = min(order, steps + 1 - step)
                        else:
                            step_order = order
                        #print('this step order:', step_order)
                        if step == steps:
                            #print('do not run corrector at the last step')
                            use_corrector = False
                        else:
                            use_corrector = True
                        x, model_x =  self.multistep_uni_pc_update(x, model_prev_list, t_prev_list, vec_t, step_order, use_corrector=use_corrector)
                        if self.after_update is not None:
                            self.after_update(x, model_x)
                        for i in range(order - 1):
                            t_prev_list[i] = t_prev_list[i + 1]
                            model_prev_list[i] = model_prev_list[i + 1]
                        t_prev_list[-1] = vec_t
                        # We do not need to evaluate the final model value.
                        if step < steps:
                            if model_x is None:
                                model_x = self.model_fn(x, vec_t)
                            model_prev_list[-1] = model_x
                        pbar.update()
        else:
            raise NotImplementedError()
        if denoise_to_zero:
            x = self.denoise_to_zero_fn(x, torch.ones((x.shape[0],)).to(device) * t_0)
        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)]