Merge pull request #440 from LuChengTHU/main

add dpm-solver support (much faster than plms)

Former-commit-id: 5a00c4f8db6c05c3e55a4d25c913796c15006e67

ldm/models/diffusion/dpm_solver/__init__.py

@@ -0,0 +1 @@
+from .sampler import DPMSolverSampler

ldm/models/diffusion/dpm_solver/dpm_solver.py

@@ -0,0 +1,1184 @@
+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.,
+        ):
+        """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("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.
+            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={},
+    guidance_type="uncond",
+    condition=None,
+    unconditional_condition=None,
+    guidance_scale=1.,
+    classifier_fn=None,
+    classifier_kwargs={},
+):
+    """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.
+    """
+
+    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, **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. 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.):
+        """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. / t_order), t_0**(1. / 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)).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 singlestep_dpm_solver_second_update(self, x, s, t, r1=0.5, model_s=None, return_intermediate=False, solver_type='dpm_solver'):
+        """
+        Singlestep solver DPM-Solver-2 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],).
+            r1: A `float`. The hyperparameter of the second-order solver.
+            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` and `s1` (the intermediate time).
+            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 solver_type not in ['dpm_solver', 'taylor']:
+            raise ValueError("'solver_type' must be either 'dpm_solver' or 'taylor', got {}".format(solver_type))
+        if r1 is None:
+            r1 = 0.5
+        ns = self.noise_schedule
+        dims = x.dim()
+        lambda_s, lambda_t = ns.marginal_lambda(s), ns.marginal_lambda(t)
+        h = lambda_t - lambda_s
+        lambda_s1 = lambda_s + r1 * h
+        s1 = ns.inverse_lambda(lambda_s1)
+        log_alpha_s, log_alpha_s1, log_alpha_t = ns.marginal_log_mean_coeff(s), ns.marginal_log_mean_coeff(s1), ns.marginal_log_mean_coeff(t)
+        sigma_s, sigma_s1, sigma_t = ns.marginal_std(s), ns.marginal_std(s1), ns.marginal_std(t)
+        alpha_s1, alpha_t = torch.exp(log_alpha_s1), torch.exp(log_alpha_t)
+
+        if self.predict_x0:
+            phi_11 = torch.expm1(-r1 * h)
+            phi_1 = torch.expm1(-h)
+
+            if model_s is None:
+                model_s = self.model_fn(x, s)
+            x_s1 = (
+                expand_dims(sigma_s1 / sigma_s, dims) * x
+                - expand_dims(alpha_s1 * phi_11, dims) * model_s
+            )
+            model_s1 = self.model_fn(x_s1, s1)
+            if solver_type == 'dpm_solver':
+                x_t = (
+                    expand_dims(sigma_t / sigma_s, dims) * x
+                    - expand_dims(alpha_t * phi_1, dims) * model_s
+                    - (0.5 / r1) * expand_dims(alpha_t * phi_1, dims) * (model_s1 - model_s)
+                )
+            elif solver_type == 'taylor':
+                x_t = (
+                    expand_dims(sigma_t / sigma_s, dims) * x
+                    - expand_dims(alpha_t * phi_1, dims) * model_s
+                    + (1. / r1) * expand_dims(alpha_t * ((torch.exp(-h) - 1.) / h + 1.), dims) * (model_s1 - model_s)
+                )
+        else:
+            phi_11 = torch.expm1(r1 * h)
+            phi_1 = torch.expm1(h)
+
+            if model_s is None:
+                model_s = self.model_fn(x, s)
+            x_s1 = (
+                expand_dims(torch.exp(log_alpha_s1 - log_alpha_s), dims) * x
+                - expand_dims(sigma_s1 * phi_11, dims) * model_s
+            )
+            model_s1 = self.model_fn(x_s1, s1)
+            if solver_type == 'dpm_solver':
+                x_t = (
+                    expand_dims(torch.exp(log_alpha_t - log_alpha_s), dims) * x
+                    - expand_dims(sigma_t * phi_1, dims) * model_s
+                    - (0.5 / r1) * expand_dims(sigma_t * phi_1, dims) * (model_s1 - model_s)
+                )
+            elif solver_type == 'taylor':
+                x_t = (
+                    expand_dims(torch.exp(log_alpha_t - log_alpha_s), dims) * x
+                    - expand_dims(sigma_t * phi_1, dims) * model_s
+                    - (1. / r1) * expand_dims(sigma_t * ((torch.exp(h) - 1.) / h - 1.), dims) * (model_s1 - model_s)
+                )
+        if return_intermediate:
+            return x_t, {'model_s': model_s, 'model_s1': model_s1}
+        else:
+            return x_t
+
+    def singlestep_dpm_solver_third_update(self, x, s, t, r1=1./3., r2=2./3., model_s=None, model_s1=None, return_intermediate=False, solver_type='dpm_solver'):
+        """
+        Singlestep solver DPM-Solver-3 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],).
+            r1: A `float`. The hyperparameter of the third-order solver.
+            r2: A `float`. The hyperparameter of the third-order solver.
+            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.
+            model_s1: A pytorch tensor. The model function evaluated at time `s1` (the intermediate time given by `r1`).
+                If `model_s1` is None, we evaluate the model at `s1`; otherwise we directly use it.
+            return_intermediate: A `bool`. If true, also return the model value at time `s`, `s1` and `s2` (the intermediate times).
+            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 solver_type not in ['dpm_solver', 'taylor']:
+            raise ValueError("'solver_type' must be either 'dpm_solver' or 'taylor', got {}".format(solver_type))
+        if r1 is None:
+            r1 = 1. / 3.
+        if r2 is None:
+            r2 = 2. / 3.
+        ns = self.noise_schedule
+        dims = x.dim()
+        lambda_s, lambda_t = ns.marginal_lambda(s), ns.marginal_lambda(t)
+        h = lambda_t - lambda_s
+        lambda_s1 = lambda_s + r1 * h
+        lambda_s2 = lambda_s + r2 * h
+        s1 = ns.inverse_lambda(lambda_s1)
+        s2 = ns.inverse_lambda(lambda_s2)
+        log_alpha_s, log_alpha_s1, log_alpha_s2, log_alpha_t = ns.marginal_log_mean_coeff(s), ns.marginal_log_mean_coeff(s1), ns.marginal_log_mean_coeff(s2), ns.marginal_log_mean_coeff(t)
+        sigma_s, sigma_s1, sigma_s2, sigma_t = ns.marginal_std(s), ns.marginal_std(s1), ns.marginal_std(s2), ns.marginal_std(t)
+        alpha_s1, alpha_s2, alpha_t = torch.exp(log_alpha_s1), torch.exp(log_alpha_s2), torch.exp(log_alpha_t)
+
+        if self.predict_x0:
+            phi_11 = torch.expm1(-r1 * h)
+            phi_12 = torch.expm1(-r2 * h)
+            phi_1 = torch.expm1(-h)
+            phi_22 = torch.expm1(-r2 * h) / (r2 * h) + 1.
+            phi_2 = phi_1 / h + 1.
+            phi_3 = phi_2 / h - 0.5
+
+            if model_s is None:
+                model_s = self.model_fn(x, s)
+            if model_s1 is None:
+                x_s1 = (
+                    expand_dims(sigma_s1 / sigma_s, dims) * x
+                    - expand_dims(alpha_s1 * phi_11, dims) * model_s
+                )
+                model_s1 = self.model_fn(x_s1, s1)
+            x_s2 = (
+                expand_dims(sigma_s2 / sigma_s, dims) * x
+                - expand_dims(alpha_s2 * phi_12, dims) * model_s
+                + r2 / r1 * expand_dims(alpha_s2 * phi_22, dims) * (model_s1 - model_s)
+            )
+            model_s2 = self.model_fn(x_s2, s2)
+            if solver_type == 'dpm_solver':
+                x_t = (
+                    expand_dims(sigma_t / sigma_s, dims) * x
+                    - expand_dims(alpha_t * phi_1, dims) * model_s
+                    + (1. / r2) * expand_dims(alpha_t * phi_2, dims) * (model_s2 - model_s)
+                )
+            elif solver_type == 'taylor':
+                D1_0 = (1. / r1) * (model_s1 - model_s)
+                D1_1 = (1. / r2) * (model_s2 - model_s)
+                D1 = (r2 * D1_0 - r1 * D1_1) / (r2 - r1)
+                D2 = 2. * (D1_1 - D1_0) / (r2 - r1)
+                x_t = (
+                    expand_dims(sigma_t / sigma_s, dims) * x
+                    - expand_dims(alpha_t * phi_1, dims) * model_s
+                    + expand_dims(alpha_t * phi_2, dims) * D1
+                    - expand_dims(alpha_t * phi_3, dims) * D2
+                )
+        else:
+            phi_11 = torch.expm1(r1 * h)
+            phi_12 = torch.expm1(r2 * h)
+            phi_1 = torch.expm1(h)
+            phi_22 = torch.expm1(r2 * h) / (r2 * h) - 1.
+            phi_2 = phi_1 / h - 1.
+            phi_3 = phi_2 / h - 0.5
+
+            if model_s is None:
+                model_s = self.model_fn(x, s)
+            if model_s1 is None:
+                x_s1 = (
+                    expand_dims(torch.exp(log_alpha_s1 - log_alpha_s), dims) * x
+                    - expand_dims(sigma_s1 * phi_11, dims) * model_s
+                )
+                model_s1 = self.model_fn(x_s1, s1)
+            x_s2 = (
+                expand_dims(torch.exp(log_alpha_s2 - log_alpha_s), dims) * x
+                - expand_dims(sigma_s2 * phi_12, dims) * model_s
+                - r2 / r1 * expand_dims(sigma_s2 * phi_22, dims) * (model_s1 - model_s)
+            )
+            model_s2 = self.model_fn(x_s2, s2)
+            if solver_type == 'dpm_solver':
+                x_t = (
+                    expand_dims(torch.exp(log_alpha_t - log_alpha_s), dims) * x
+                    - expand_dims(sigma_t * phi_1, dims) * model_s
+                    - (1. / r2) * expand_dims(sigma_t * phi_2, dims) * (model_s2 - model_s)
+                )
+            elif solver_type == 'taylor':
+                D1_0 = (1. / r1) * (model_s1 - model_s)
+                D1_1 = (1. / r2) * (model_s2 - model_s)
+                D1 = (r2 * D1_0 - r1 * D1_1) / (r2 - r1)
+                D2 = 2. * (D1_1 - D1_0) / (r2 - r1)
+                x_t = (
+                    expand_dims(torch.exp(log_alpha_t - log_alpha_s), dims) * x
+                    - expand_dims(sigma_t * phi_1, dims) * model_s
+                    - expand_dims(sigma_t * phi_2, dims) * D1
+                    - expand_dims(sigma_t * phi_3, dims) * D2
+                )
+
+        if return_intermediate:
+            return x_t, {'model_s': model_s, 'model_s1': model_s1, 'model_s2': model_s2}
+        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`.
+        """
+        if solver_type not in ['dpm_solver', 'taylor']:
+            raise ValueError("'solver_type' must be either 'dpm_solver' or 'taylor', got {}".format(solver_type))
+        ns = self.noise_schedule
+        dims = x.dim()
+        model_prev_1, model_prev_0 = model_prev_list
+        t_prev_1, t_prev_0 = t_prev_list
+        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. / r0, dims) * (model_prev_0 - model_prev_1)
+        if self.predict_x0:
+            if solver_type == 'dpm_solver':
+                x_t = (
+                    expand_dims(sigma_t / sigma_prev_0, dims) * x
+                    - expand_dims(alpha_t * (torch.exp(-h) - 1.), dims) * model_prev_0
+                    - 0.5 * expand_dims(alpha_t * (torch.exp(-h) - 1.), 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.), dims) * model_prev_0
+                    + expand_dims(alpha_t * ((torch.exp(-h) - 1.) / h + 1.), dims) * D1_0
+                )
+        else:
+            if solver_type == 'dpm_solver':
+                x_t = (
+                    expand_dims(torch.exp(log_alpha_t - log_alpha_prev_0), dims) * x
+                    - expand_dims(sigma_t * (torch.exp(h) - 1.), dims) * model_prev_0
+                    - 0.5 * expand_dims(sigma_t * (torch.exp(h) - 1.), 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.), dims) * model_prev_0
+                    - expand_dims(sigma_t * ((torch.exp(h) - 1.) / h - 1.), 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. / r0, dims) * (model_prev_0 - model_prev_1)
+        D1_1 = expand_dims(1. / 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. / (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.), dims) * model_prev_0
+                + expand_dims(alpha_t * ((torch.exp(-h) - 1.) / h + 1.), dims) * D1
+                - expand_dims(alpha_t * ((torch.exp(-h) - 1. + 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.), dims) * model_prev_0
+                - expand_dims(sigma_t * ((torch.exp(h) - 1.) / h - 1.), dims) * D1
+                - expand_dims(sigma_t * ((torch.exp(h) - 1. - h) / h**2 - 0.5), dims) * D2
+            )
+        return x_t
+
+    def singlestep_dpm_solver_update(self, x, s, t, order, return_intermediate=False, solver_type='dpm_solver', r1=None, r2=None):
+        """
+        Singlestep DPM-Solver with the order `order` 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],).
+            order: A `int`. The order of DPM-Solver. We only support order == 1 or 2 or 3.
+            return_intermediate: A `bool`. If true, also return the model value at time `s`, `s1` and `s2` (the intermediate times).
+            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.
+            r1: A `float`. The hyperparameter of the second-order or third-order solver.
+            r2: A `float`. The hyperparameter of the third-order solver.
+        Returns:
+            x_t: A pytorch tensor. The approximated solution at time `t`.
+        """
+        if order == 1:
+            return self.dpm_solver_first_update(x, s, t, return_intermediate=return_intermediate)
+        elif order == 2:
+            return self.singlestep_dpm_solver_second_update(x, s, t, return_intermediate=return_intermediate, solver_type=solver_type, r1=r1)
+        elif order == 3:
+            return self.singlestep_dpm_solver_third_update(x, s, t, return_intermediate=return_intermediate, solver_type=solver_type, r1=r1, r2=r2)
+        else:
+            raise ValueError("Solver order must be 1 or 2 or 3, got {}".format(order))
+
+    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. / 3., 2. / 3.
+            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.):
+                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. / 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,
+    ):
+        """
+        Compute the sample at time `t_end` by DPM-Solver, given the initial `x` at time `t_start`.
+
+        =====================================================
+
+        We support the following algorithms for both noise prediction model and data prediction model:
+            - 'singlestep':
+                Singlestep DPM-Solver (i.e. "DPM-Solver-fast" in the paper), which combines different orders of singlestep DPM-Solver. 
+                We combine all the singlestep solvers with order <= `order` to use up all the function evaluations (steps).
+                The total number of function evaluations (NFE) == `steps`.
+                Given a fixed NFE == `steps`, the sampling procedure is:
+                    - If `order` == 1:
+                        - Denote K = steps. We use K steps of DPM-Solver-1 (i.e. DDIM).
+                    - If `order` == 2:
+                        - Denote K = (steps // 2) + (steps % 2). We take K intermediate time steps for sampling.
+                        - If steps % 2 == 0, we use K steps of singlestep DPM-Solver-2.
+                        - If steps % 2 == 1, we use (K - 1) steps of singlestep 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 singlestep DPM-Solver-3, and 1 step of singlestep DPM-Solver-2 and 1 step of DPM-Solver-1.
+                        - If steps % 3 == 1, we use (K - 1) steps of singlestep DPM-Solver-3 and 1 step of DPM-Solver-1.
+                        - If steps % 3 == 2, we use (K - 1) steps of singlestep DPM-Solver-3 and 1 step of singlestep DPM-Solver-2.
+            - 'multistep':
+                Multistep DPM-Solver with the order of `order`. The total number of function evaluations (NFE) == `steps`.
+                We initialize the first `order` values by lower order multistep solvers.
+                Given a fixed NFE == `steps`, the sampling procedure is:
+                    Denote K = steps.
+                    - If `order` == 1:
+                        - We use K steps of DPM-Solver-1 (i.e. DDIM).
+                    - If `order` == 2:
+                        - We firstly use 1 step of DPM-Solver-1, then use (K - 1) step of multistep DPM-Solver-2.
+                    - If `order` == 3:
+                        - We firstly use 1 step of DPM-Solver-1, then 1 step of multistep DPM-Solver-2, then (K - 2) step of multistep DPM-Solver-3.
+            - 'singlestep_fixed':
+                Fixed order singlestep DPM-Solver (i.e. DPM-Solver-1 or singlestep DPM-Solver-2 or singlestep DPM-Solver-3).
+                We use singlestep DPM-Solver-`order` for `order`=1 or 2 or 3, with total [`steps` // `order`] * `order` NFE.
+            - 'adaptive':
+                Adaptive step size DPM-Solver (i.e. "DPM-Solver-12" and "DPM-Solver-23" in the paper).
+                We ignore `steps` and use adaptive step size DPM-Solver with a higher order of `order`.
+                You can adjust the absolute tolerance `atol` and the relative tolerance `rtol` to balance the computatation costs
+                (NFE) and the sample quality.
+                    - If `order` == 2, we use DPM-Solver-12 which combines DPM-Solver-1 and singlestep DPM-Solver-2.
+                    - If `order` == 3, we use DPM-Solver-23 which combines singlestep DPM-Solver-2 and singlestep DPM-Solver-3.
+
+        =====================================================
+
+        Some advices for choosing the algorithm:
+            - For **unconditional sampling** or **guided sampling with small guidance scale** by DPMs:
+                Use singlestep DPM-Solver ("DPM-Solver-fast" in the paper) with `order = 3`.
+                e.g.
+                    >>> dpm_solver = DPM_Solver(model_fn, noise_schedule, predict_x0=False)
+                    >>> x_sample = dpm_solver.sample(x, steps=steps, t_start=t_start, t_end=t_end, order=3,
+                            skip_type='time_uniform', method='singlestep')
+            - For **guided sampling with large guidance scale** by DPMs:
+                Use multistep DPM-Solver with `predict_x0 = True` and `order = 2`.
+                e.g.
+                    >>> dpm_solver = DPM_Solver(model_fn, noise_schedule, predict_x0=True)
+                    >>> x_sample = dpm_solver.sample(x, steps=steps, t_start=t_start, t_end=t_end, order=2,
+                            skip_type='time_uniform', method='multistep')
+
+        We support three types of `skip_type`:
+            - 'logSNR': uniform logSNR for the time steps. **Recommended for low-resolutional images**
+            - 'time_uniform': uniform time for the time steps. **Recommended for high-resolutional images**.
+            - 'time_quadratic': quadratic time for the time steps.
+
+        =====================================================
+        Args:
+            x: A pytorch tensor. The initial value at time `t_start`
+                e.g. if `t_start` == T, then `x` is a sample from the standard normal distribution.
+            steps: A `int`. The total number of function evaluations (NFE).
+            t_start: A `float`. The starting time of the sampling.
+                If `T` is None, we use self.noise_schedule.T (default is 1.0).
+            t_end: A `float`. The ending time of the sampling.
+                If `t_end` is None, we use 1. / self.noise_schedule.total_N.
+                e.g. if total_N == 1000, we have `t_end` == 1e-3.
+                For discrete-time DPMs:
+                    - We recommend `t_end` == 1. / self.noise_schedule.total_N.
+                For continuous-time DPMs:
+                    - We recommend `t_end` == 1e-3 when `steps` <= 15; and `t_end` == 1e-4 when `steps` > 15.
+            order: A `int`. The order of DPM-Solver.
+            skip_type: A `str`. The type for the spacing of the time steps. 'time_uniform' or 'logSNR' or 'time_quadratic'.
+            method: A `str`. The method for sampling. 'singlestep' or 'multistep' or 'singlestep_fixed' or 'adaptive'.
+            denoise_to_zero: A `bool`. Whether to denoise to time 0 at the final step.
+                Default is `False`. If `denoise_to_zero` is `True`, the total NFE is (`steps` + 1).
+
+                This trick is firstly proposed by DDPM (https://arxiv.org/abs/2006.11239) and
+                score_sde (https://arxiv.org/abs/2011.13456). Such trick can improve the FID
+                for diffusion models sampling by diffusion SDEs for low-resolutional images
+                (such as CIFAR-10). However, we observed that such trick does not matter for
+                high-resolutional images. As it needs an additional NFE, we do not recommend
+                it for high-resolutional images.
+            lower_order_final: A `bool`. Whether to use lower order solvers at the final steps.
+                Only valid for `method=multistep` and `steps < 15`. We empirically find that
+                this trick is a key to stabilizing the sampling by DPM-Solver with very few steps
+                (especially for steps <= 10). So we recommend to set it to be `True`.
+            solver_type: A `str`. The taylor expansion type for the solver. `dpm_solver` or `taylor`. We recommend `dpm_solver`.
+            atol: A `float`. The absolute tolerance of the adaptive step size solver. Valid when `method` == 'adaptive'.
+            rtol: A `float`. The relative tolerance of the adaptive step size solver. Valid when `method` == 'adaptive'.
+        Returns:
+            x_end: A pytorch tensor. The approximated solution at time `t_end`.
+
+        """
+        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 == 'adaptive':
+            with torch.no_grad():
+                x = self.dpm_solver_adaptive(x, order=order, t_T=t_T, t_0=t_0, atol=atol, rtol=rtol, solver_type=solver_type)
+        elif method == 'multistep':
+            assert steps >= order
+            timesteps = self.get_time_steps(skip_type=skip_type, t_T=t_T, t_0=t_0, N=steps, device=device)
+            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]
+                # 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 = self.multistep_dpm_solver_update(x, model_prev_list, t_prev_list, vec_t, init_order, solver_type=solver_type)
+                    model_prev_list.append(self.model_fn(x, vec_t))
+                    t_prev_list.append(vec_t)
+                # Compute the remaining values by `order`-th order multistep DPM-Solver.
+                for step in range(order, steps + 1):
+                    vec_t = timesteps[step].expand(x.shape[0])
+                    if lower_order_final and steps < 15:
+                        step_order = min(order, steps + 1 - step)
+                    else:
+                        step_order = order
+                    x = self.multistep_dpm_solver_update(x, model_prev_list, t_prev_list, vec_t, step_order, solver_type=solver_type)
+                    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:
+                        model_prev_list[-1] = self.model_fn(x, vec_t)
+        elif method in ['singlestep', 'singlestep_fixed']:
+            if method == 'singlestep':
+                timesteps_outer, orders = self.get_orders_and_timesteps_for_singlestep_solver(steps=steps, order=order, skip_type=skip_type, t_T=t_T, t_0=t_0, device=device)
+            elif method == 'singlestep_fixed':
+                K = steps // order
+                orders = [order,] * K
+                timesteps_outer = self.get_time_steps(skip_type=skip_type, t_T=t_T, t_0=t_0, N=K, device=device)
+            for i, order in enumerate(orders):
+                t_T_inner, t_0_inner = timesteps_outer[i], timesteps_outer[i + 1]
+                timesteps_inner = self.get_time_steps(skip_type=skip_type, t_T=t_T_inner.item(), t_0=t_0_inner.item(), N=order, device=device)
+                lambda_inner = self.noise_schedule.marginal_lambda(timesteps_inner)
+                vec_s, vec_t = t_T_inner.tile(x.shape[0]), t_0_inner.tile(x.shape[0])
+                h = lambda_inner[-1] - lambda_inner[0]
+                r1 = None if order <= 1 else (lambda_inner[1] - lambda_inner[0]) / h
+                r2 = None if order <= 2 else (lambda_inner[2] - lambda_inner[0]) / h
+                x = self.singlestep_dpm_solver_update(x, vec_s, vec_t, order, solver_type=solver_type, r1=r1, r2=r2)
+        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)]

ldm/models/diffusion/dpm_solver/sampler.py

@@ -0,0 +1,82 @@
+"""SAMPLING ONLY."""
+
+import torch
+
+from .solver import NoiseScheduleVP, model_wrapper, DPM_Solver
+
+
+class DPMSolverSampler(object):
+    def __init__(self, model, **kwargs):
+        super().__init__()
+        self.model = model
+        to_torch = lambda x: x.clone().detach().to(torch.float32).to(model.device)
+        self.register_buffer('alphas_cumprod', to_torch(model.alphas_cumprod))
+
+    def register_buffer(self, name, attr):
+        if type(attr) == torch.Tensor:
+            if attr.device != torch.device("cuda"):
+                attr = attr.to(torch.device("cuda"))
+        setattr(self, name, attr)
+
+    @torch.no_grad()
+    def sample(self,
+               S,
+               batch_size,
+               shape,
+               conditioning=None,
+               callback=None,
+               normals_sequence=None,
+               img_callback=None,
+               quantize_x0=False,
+               eta=0.,
+               mask=None,
+               x0=None,
+               temperature=1.,
+               noise_dropout=0.,
+               score_corrector=None,
+               corrector_kwargs=None,
+               verbose=True,
+               x_T=None,
+               log_every_t=100,
+               unconditional_guidance_scale=1.,
+               unconditional_conditioning=None,
+               # this has to come in the same format as the conditioning, # e.g. as encoded tokens, ...
+               **kwargs
+               ):
+        if conditioning is not None:
+            if isinstance(conditioning, dict):
+                cbs = conditioning[list(conditioning.keys())[0]].shape[0]
+                if cbs != batch_size:
+                    print(f"Warning: Got {cbs} conditionings but batch-size is {batch_size}")
+            else:
+                if conditioning.shape[0] != batch_size:
+                    print(f"Warning: Got {conditioning.shape[0]} conditionings but batch-size is {batch_size}")
+
+        # sampling
+        C, H, W = shape
+        size = (batch_size, C, H, W)
+
+        # print(f'Data shape for DPM-Solver sampling is {size}, sampling steps {S}')
+
+        device = self.model.betas.device
+        if x_T is None:
+            img = torch.randn(size, device=device)
+        else:
+            img = x_T
+
+        ns = NoiseScheduleVP('discrete', alphas_cumprod=self.alphas_cumprod)
+
+        model_fn = model_wrapper(
+            lambda x, t, c: self.model.apply_model(x, t, c),
+            ns,
+            model_type="noise",
+            guidance_type="classifier-free",
+            condition=conditioning,
+            unconditional_condition=unconditional_conditioning,
+            guidance_scale=unconditional_guidance_scale,
+        )
+
+        dpm_solver = DPM_Solver(model_fn, ns, predict_x0=True, thresholding=False)
+        x = dpm_solver.sample(img, steps=S, skip_type="time_uniform", method="multistep", order=2, lower_order_final=True)
+
+        return x.to(device), None

scripts/txt2img.py

@@ -17,6 +17,7 @@ from contextlib import contextmanager, nullcontext
 from ldm.util import instantiate_from_config
 from ldm.models.diffusion.ddim import DDIMSampler
 from ldm.models.diffusion.plms import PLMSSampler
+from ldm.models.diffusion.dpm_solver import DPMSolverSampler
 
 from diffusers.pipelines.stable_diffusion.safety_checker import StableDiffusionSafetyChecker
 from transformers import AutoFeatureExtractor
@@ -132,6 +133,11 @@ def main():
         action='store_true',
         help="use plms sampling",
     )
+    parser.add_argument(
+        "--dpm_solver",
+        action='store_true',
+        help="use dpm_solver sampling",
+    )
     parser.add_argument(
         "--laion400m",
         action='store_true',
@@ -242,7 +248,9 @@ def main():
     device = torch.device("cuda") if torch.cuda.is_available() else torch.device("cpu")
     model = model.to(device)
 
-    if opt.plms:
+    if opt.dpm_solver:
+        sampler = DPMSolverSampler(model)
+    elif opt.plms:
         sampler = PLMSSampler(model)
     else:
         sampler = DDIMSampler(model)