import math import torch class NoiseScheduleVP: def __init__( self, schedule="discrete", betas=None, alphas_cumprod=None, continuous_beta_0=0.1, continuous_beta_1=20.0, dtype=torch.float32, ): """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(1 - 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 = 1.0 self.t_array = torch.linspace(0.0, 1.0, self.total_N + 1)[1:].reshape((1, -1)).to(dtype=dtype) self.log_alpha_array = log_alphas.reshape( ( 1, -1, ) ).to(dtype=dtype) else: self.total_N = 1000 self.beta_0 = continuous_beta_0 self.beta_1 = continuous_beta_1 self.cosine_s = 0.008 self.cosine_beta_max = 999.0 self.cosine_t_max = ( math.atan(self.cosine_beta_max * (1.0 + self.cosine_s) / math.pi) * 2.0 * (1.0 + self.cosine_s) / math.pi - self.cosine_s ) self.cosine_log_alpha_0 = math.log(math.cos(self.cosine_s / (1.0 + self.cosine_s) * math.pi / 2.0)) self.schedule = schedule if schedule == "cosine": # For the cosine schedule, T = 1 will have numerical issues. So we manually set the ending time T. # Note that T = 0.9946 may be not the optimal setting. However, we find it works well. self.T = 0.9946 else: self.T = 1.0 def marginal_log_mean_coeff(self, t): """ Compute log(alpha_t) of a given continuous-time label t in [0, T]. """ if self.schedule == "discrete": return interpolate_fn( t.reshape((-1, 1)), self.t_array.to(t.device), self.log_alpha_array.to(t.device), ).reshape((-1)) elif self.schedule == "linear": return -0.25 * t**2 * (self.beta_1 - self.beta_0) - 0.5 * t * self.beta_0 elif self.schedule == "cosine": def log_alpha_fn(s): return torch.log(torch.cos((s + self.cosine_s) / (1.0 + self.cosine_s) * math.pi / 2.0)) log_alpha_t = log_alpha_fn(t) - self.cosine_log_alpha_0 return log_alpha_t def marginal_alpha(self, t): """ Compute alpha_t of a given continuous-time label t in [0, T]. """ return torch.exp(self.marginal_log_mean_coeff(t)) def marginal_std(self, t): """ Compute sigma_t of a given continuous-time label t in [0, T]. """ return torch.sqrt(1.0 - torch.exp(2.0 * self.marginal_log_mean_coeff(t))) def marginal_lambda(self, t): """ Compute lambda_t = log(alpha_t) - log(sigma_t) of a given continuous-time label t in [0, T]. """ log_mean_coeff = self.marginal_log_mean_coeff(t) log_std = 0.5 * torch.log(1.0 - torch.exp(2.0 * log_mean_coeff)) return log_mean_coeff - log_std def inverse_lambda(self, lamb): """ Compute the continuous-time label t in [0, T] of a given half-logSNR lambda_t. """ if self.schedule == "linear": tmp = 2.0 * (self.beta_1 - self.beta_0) * torch.logaddexp(-2.0 * lamb, torch.zeros((1,)).to(lamb)) Delta = self.beta_0**2 + tmp return tmp / (torch.sqrt(Delta) + self.beta_0) / (self.beta_1 - self.beta_0) elif self.schedule == "discrete": log_alpha = -0.5 * torch.logaddexp(torch.zeros((1,)).to(lamb.device), -2.0 * lamb) t = interpolate_fn( log_alpha.reshape((-1, 1)), torch.flip(self.log_alpha_array.to(lamb.device), [1]), torch.flip(self.t_array.to(lamb.device), [1]), ) return t.reshape((-1,)) else: log_alpha = -0.5 * torch.logaddexp(-2.0 * lamb, torch.zeros((1,)).to(lamb)) def t_fn(log_alpha_t): return ( torch.arccos(torch.exp(log_alpha_t + self.cosine_log_alpha_0)) * 2.0 * (1.0 + self.cosine_s) / math.pi - self.cosine_s ) t = t_fn(log_alpha) return t def model_wrapper( model, noise_schedule, model_type="noise", model_kwargs={}, guidance_type="uncond", condition=None, unconditional_condition=None, guidance_scale=1.0, classifier_fn=None, classifier_kwargs={}, ): """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 - 1.0 / noise_schedule.total_N) * 1000.0 else: return t_continuous def noise_pred_fn(x, t_continuous, cond=None): 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) return (x - alpha_t * output) / sigma_t elif model_type == "v": alpha_t, sigma_t = noise_schedule.marginal_alpha(t_continuous), noise_schedule.marginal_std(t_continuous) return alpha_t * output + sigma_t * x elif model_type == "score": sigma_t = noise_schedule.marginal_std(t_continuous) return -sigma_t * 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 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 * sigma_t * cond_grad elif guidance_type == "classifier-free": if guidance_scale == 1.0 or unconditional_condition is None: return noise_pred_fn(x, t_continuous, cond=condition) else: x_in = torch.cat([x] * 2) t_in = torch.cat([t_continuous] * 2) c_in = torch.cat([unconditional_condition, condition]) noise_uncond, noise = noise_pred_fn(x_in, t_in, cond=c_in).chunk(2) return noise_uncond + guidance_scale * (noise - noise_uncond) assert model_type in ["noise", "x_start", "v", "score"] assert guidance_type in ["uncond", "classifier", "classifier-free"] return model_fn class DPM_Solver: def __init__( self, model_fn, noise_schedule, algorithm_type="dpmsolver++", correcting_x0_fn=None, correcting_xt_fn=None, thresholding_max_val=1.0, dynamic_thresholding_ratio=0.995, ): """Construct a DPM-Solver. We support both DPM-Solver (`algorithm_type="dpmsolver"`) and DPM-Solver++ (`algorithm_type="dpmsolver++"`). We also support the "dynamic thresholding" method in Imagen[1]. For pixel-space diffusion models, you can set both `algorithm_type="dpmsolver++"` and `correcting_x0_fn="dynamic_thresholding"` to use the dynamic thresholding. The "dynamic thresholding" can greatly improve the sample quality for pixel-space DPMs with large guidance scales. Note that the thresholding method is **unsuitable** for latent-space DPMs (such as stable-diffusion). To support advanced algorithms in image-to-image applications, we also support corrector functions for both x0 and xt. 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 `` The shape of `x` is `(batch_size, **shape)`, and the shape of `t_continuous` is `(batch_size,)`. noise_schedule: A noise schedule object, such as NoiseScheduleVP. algorithm_type: A `str`. Either "dpmsolver" or "dpmsolver++". correcting_x0_fn: A `str` or a function with the following format: ``` def correcting_x0_fn(x0, t): x0_new = ... return x0_new ``` This function is to correct the outputs of the data prediction model at each sampling step. e.g., ``` x0_pred = data_pred_model(xt, t) if correcting_x0_fn is not None: x0_pred = correcting_x0_fn(x0_pred, t) xt_1 = update(x0_pred, xt, t) ``` If `correcting_x0_fn="dynamic_thresholding"`, we use the dynamic thresholding proposed in Imagen[1]. correcting_xt_fn: A function with the following format: ``` def correcting_xt_fn(xt, t, step): x_new = ... return x_new ``` This function is to correct the intermediate samples xt at each sampling step. e.g., ``` xt = ... xt = correcting_xt_fn(xt, t, step) ``` thresholding_max_val: A `float`. The max value for thresholding. Valid only when use `dpmsolver++` and `correcting_x0_fn="dynamic_thresholding"`. dynamic_thresholding_ratio: A `float`. The ratio for dynamic thresholding (see Imagen[1] for details). Valid only when use `dpmsolver++` and `correcting_x0_fn="dynamic_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 = lambda x, t: model_fn(x, t.expand((x.shape[0]))) self.noise_schedule = noise_schedule assert algorithm_type in ["dpmsolver", "dpmsolver++"] self.algorithm_type = algorithm_type if correcting_x0_fn == "dynamic_thresholding": self.correcting_x0_fn = self.dynamic_thresholding_fn else: self.correcting_x0_fn = correcting_x0_fn self.correcting_xt_fn = correcting_xt_fn self.dynamic_thresholding_ratio = dynamic_thresholding_ratio self.thresholding_max_val = thresholding_max_val def dynamic_thresholding_fn(self, x0, t): """ 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 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 corrector). """ noise = self.noise_prediction_fn(x, t) alpha_t, sigma_t = self.noise_schedule.marginal_alpha(t), self.noise_schedule.marginal_std(t) x0 = (x - sigma_t * noise) / alpha_t if self.correcting_x0_fn is not None: x0 = self.correcting_x0_fn(x0, t) return x0 def model_fn(self, x, t): """ Convert the model to the noise prediction model or the data prediction model. """ if self.algorithm_type == "dpmsolver++": return self.data_prediction_fn(x, t) else: return self.noise_prediction_fn(x, t) def get_time_steps(self, skip_type, t_T, t_0, N, device): """Compute the intermediate time steps for sampling. Args: skip_type: A `str`. The type for the spacing of the time steps. We support three types: - 'logSNR': uniform logSNR for the time steps. - 'time_uniform': uniform time for the time steps. (**Recommended for high-resolutional data**.) - 'time_quadratic': quadratic time for the time steps. (Used in DDIM for low-resolutional data.) t_T: A `float`. The starting time of the sampling (default is T). t_0: A `float`. The ending time of the sampling (default is epsilon). N: A `int`. The total number of the spacing of the time steps. device: A torch device. Returns: A pytorch tensor of the time steps, with the shape (N + 1,). """ if skip_type == "logSNR": lambda_T = self.noise_schedule.marginal_lambda(torch.tensor(t_T).to(device)) lambda_0 = self.noise_schedule.marginal_lambda(torch.tensor(t_0).to(device)) logSNR_steps = torch.linspace(lambda_T.cpu().item(), lambda_0.cpu().item(), N + 1).to(device) return self.noise_schedule.inverse_lambda(logSNR_steps) elif skip_type == "time_uniform": return torch.linspace(t_T, t_0, N + 1).to(device) elif skip_type == "time_quadratic": t_order = 2 t = torch.linspace(t_T ** (1.0 / t_order), t_0 ** (1.0 / t_order), N + 1).pow(t_order).to(device) return t else: raise ValueError( "Unsupported skip_type {}, need to be 'logSNR' or 'time_uniform' or 'time_quadratic'".format(skip_type) ) def get_orders_and_timesteps_for_singlestep_solver(self, steps, order, skip_type, t_T, t_0, device): """ Get the order of each step for sampling by the singlestep DPM-Solver. We combine both DPM-Solver-1,2,3 to use all the function evaluations, which is named as "DPM-Solver-fast". Given a fixed number of function evaluations by `steps`, the sampling procedure by DPM-Solver-fast is: - If order == 1: We take `steps` of DPM-Solver-1 (i.e. DDIM). - If order == 2: - Denote K = (steps // 2). We take K or (K + 1) intermediate time steps for sampling. - If steps % 2 == 0, we use K steps of DPM-Solver-2. - If steps % 2 == 1, we use K steps of DPM-Solver-2 and 1 step of DPM-Solver-1. - If order == 3: - Denote K = (steps // 3 + 1). We take K intermediate time steps for sampling. - If steps % 3 == 0, we use (K - 2) steps of DPM-Solver-3, and 1 step of DPM-Solver-2 and 1 step of DPM-Solver-1. - If steps % 3 == 1, we use (K - 1) steps of DPM-Solver-3 and 1 step of DPM-Solver-1. - If steps % 3 == 2, we use (K - 1) steps of DPM-Solver-3 and 1 step of DPM-Solver-2. ============================================ Args: order: A `int`. The max order for the solver (2 or 3). steps: A `int`. The total number of function evaluations (NFE). skip_type: A `str`. The type for the spacing of the time steps. We support three types: - 'logSNR': uniform logSNR for the time steps. - 'time_uniform': uniform time for the time steps. (**Recommended for high-resolutional data**.) - 'time_quadratic': quadratic time for the time steps. (Used in DDIM for low-resolutional data.) t_T: A `float`. The starting time of the sampling (default is T). t_0: A `float`. The ending time of the sampling (default is epsilon). device: A torch device. Returns: orders: A list of the solver order of each step. """ if order == 3: K = steps // 3 + 1 if steps % 3 == 0: orders = [ 3, ] * ( K - 2 ) + [2, 1] elif steps % 3 == 1: orders = [ 3, ] * ( K - 1 ) + [1] else: orders = [ 3, ] * ( K - 1 ) + [2] elif order == 2: if steps % 2 == 0: K = steps // 2 orders = [ 2, ] * K else: K = steps // 2 + 1 orders = [ 2, ] * ( K - 1 ) + [1] elif order == 1: K = 1 orders = [ 1, ] * steps else: raise ValueError("'order' must be '1' or '2' or '3'.") if skip_type == "logSNR": # To reproduce the results in DPM-Solver paper timesteps_outer = self.get_time_steps(skip_type, t_T, t_0, K, device) else: timesteps_outer = self.get_time_steps(skip_type, t_T, t_0, steps, device)[ torch.cumsum( torch.tensor( [ 0, ] + orders ), 0, ).to(device) ] return timesteps_outer, orders def denoise_to_zero_fn(self, x, s): """ Denoise at the final step, which is equivalent to solve the ODE from lambda_s to infty by first-order discretization. """ return self.data_prediction_fn(x, s) def dpm_solver_first_update(self, x, s, t, model_s=None, return_intermediate=False): """ DPM-Solver-1 (equivalent to DDIM) from time `s` to time `t`. Args: x: A pytorch tensor. The initial value at time `s`. s: A pytorch tensor. The starting time, with the shape (1,). t: A pytorch tensor. The ending time, with the shape (1,). 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.algorithm_type == "dpmsolver++": phi_1 = torch.expm1(-h) if model_s is None: model_s = self.model_fn(x, s) x_t = sigma_t / sigma_s * x - alpha_t * phi_1 * 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 = torch.exp(log_alpha_t - log_alpha_s) * x - (sigma_t * phi_1) * 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="dpmsolver", ): """ 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 (1,). t: A pytorch tensor. The ending time, with the shape (1,). 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 'dpmsolver' or 'taylor'. The type for the high-order solvers. The type slightly impacts the performance. We recommend to use 'dpmsolver' type. Returns: x_t: A pytorch tensor. The approximated solution at time `t`. """ if solver_type not in ["dpmsolver", "taylor"]: raise ValueError("'solver_type' must be either 'dpmsolver' or 'taylor', got {}".format(solver_type)) if r1 is None: r1 = 0.5 ns = self.noise_schedule 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.algorithm_type == "dpmsolver++": 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 = (sigma_s1 / sigma_s) * x - (alpha_s1 * phi_11) * model_s model_s1 = self.model_fn(x_s1, s1) if solver_type == "dpmsolver": x_t = ( (sigma_t / sigma_s) * x - (alpha_t * phi_1) * model_s - (0.5 / r1) * (alpha_t * phi_1) * (model_s1 - model_s) ) elif solver_type == "taylor": x_t = ( (sigma_t / sigma_s) * x - (alpha_t * phi_1) * model_s + (1.0 / r1) * (alpha_t * (phi_1 / h + 1.0)) * (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 = torch.exp(log_alpha_s1 - log_alpha_s) * x - (sigma_s1 * phi_11) * model_s model_s1 = self.model_fn(x_s1, s1) if solver_type == "dpmsolver": x_t = ( torch.exp(log_alpha_t - log_alpha_s) * x - (sigma_t * phi_1) * model_s - (0.5 / r1) * (sigma_t * phi_1) * (model_s1 - model_s) ) elif solver_type == "taylor": x_t = ( torch.exp(log_alpha_t - log_alpha_s) * x - (sigma_t * phi_1) * model_s - (1.0 / r1) * (sigma_t * (phi_1 / h - 1.0)) * (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=1.0 / 3.0, r2=2.0 / 3.0, model_s=None, model_s1=None, return_intermediate=False, solver_type="dpmsolver", ): """ 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 (1,). t: A pytorch tensor. The ending time, with the shape (1,). 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 'dpmsolver' or 'taylor'. The type for the high-order solvers. The type slightly impacts the performance. We recommend to use 'dpmsolver' type. Returns: x_t: A pytorch tensor. The approximated solution at time `t`. """ if solver_type not in ["dpmsolver", "taylor"]: raise ValueError("'solver_type' must be either 'dpmsolver' or 'taylor', got {}".format(solver_type)) if r1 is None: r1 = 1.0 / 3.0 if r2 is None: r2 = 2.0 / 3.0 ns = self.noise_schedule 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.algorithm_type == "dpmsolver++": 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) + 1.0 phi_2 = phi_1 / h + 1.0 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 = (sigma_s1 / sigma_s) * x - (alpha_s1 * phi_11) * model_s model_s1 = self.model_fn(x_s1, s1) x_s2 = ( (sigma_s2 / sigma_s) * x - (alpha_s2 * phi_12) * model_s + r2 / r1 * (alpha_s2 * phi_22) * (model_s1 - model_s) ) model_s2 = self.model_fn(x_s2, s2) if solver_type == "dpmsolver": x_t = ( (sigma_t / sigma_s) * x - (alpha_t * phi_1) * model_s + (1.0 / r2) * (alpha_t * phi_2) * (model_s2 - model_s) ) elif solver_type == "taylor": D1_0 = (1.0 / r1) * (model_s1 - model_s) D1_1 = (1.0 / r2) * (model_s2 - model_s) D1 = (r2 * D1_0 - r1 * D1_1) / (r2 - r1) D2 = 2.0 * (D1_1 - D1_0) / (r2 - r1) x_t = ( (sigma_t / sigma_s) * x - (alpha_t * phi_1) * model_s + (alpha_t * phi_2) * D1 - (alpha_t * phi_3) * 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) - 1.0 phi_2 = phi_1 / h - 1.0 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 = (torch.exp(log_alpha_s1 - log_alpha_s)) * x - (sigma_s1 * phi_11) * model_s model_s1 = self.model_fn(x_s1, s1) x_s2 = ( (torch.exp(log_alpha_s2 - log_alpha_s)) * x - (sigma_s2 * phi_12) * model_s - r2 / r1 * (sigma_s2 * phi_22) * (model_s1 - model_s) ) model_s2 = self.model_fn(x_s2, s2) if solver_type == "dpmsolver": x_t = ( (torch.exp(log_alpha_t - log_alpha_s)) * x - (sigma_t * phi_1) * model_s - (1.0 / r2) * (sigma_t * phi_2) * (model_s2 - model_s) ) elif solver_type == "taylor": D1_0 = (1.0 / r1) * (model_s1 - model_s) D1_1 = (1.0 / r2) * (model_s2 - model_s) D1 = (r2 * D1_0 - r1 * D1_1) / (r2 - r1) D2 = 2.0 * (D1_1 - D1_0) / (r2 - r1) x_t = ( (torch.exp(log_alpha_t - log_alpha_s)) * x - (sigma_t * phi_1) * model_s - (sigma_t * phi_2) * D1 - (sigma_t * phi_3) * 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="dpmsolver"): """ 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 (1,) t: A pytorch tensor. The ending time, with the shape (1,). solver_type: either 'dpmsolver' or 'taylor'. The type for the high-order solvers. The type slightly impacts the performance. We recommend to use 'dpmsolver' type. Returns: x_t: A pytorch tensor. The approximated solution at time `t`. """ if solver_type not in ["dpmsolver", "taylor"]: raise ValueError("'solver_type' must be either 'dpmsolver' or 'taylor', got {}".format(solver_type)) ns = self.noise_schedule model_prev_1, model_prev_0 = model_prev_list[-2], model_prev_list[-1] t_prev_1, t_prev_0 = t_prev_list[-2], t_prev_list[-1] 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 = (1.0 / r0) * (model_prev_0 - model_prev_1) if self.algorithm_type == "dpmsolver++": phi_1 = torch.expm1(-h) if solver_type == "dpmsolver": x_t = (sigma_t / sigma_prev_0) * x - (alpha_t * phi_1) * model_prev_0 - 0.5 * (alpha_t * phi_1) * D1_0 elif solver_type == "taylor": x_t = ( (sigma_t / sigma_prev_0) * x - (alpha_t * phi_1) * model_prev_0 + (alpha_t * (phi_1 / h + 1.0)) * D1_0 ) else: phi_1 = torch.expm1(h) if solver_type == "dpmsolver": x_t = ( (torch.exp(log_alpha_t - log_alpha_prev_0)) * x - (sigma_t * phi_1) * model_prev_0 - 0.5 * (sigma_t * phi_1) * D1_0 ) elif solver_type == "taylor": x_t = ( (torch.exp(log_alpha_t - log_alpha_prev_0)) * x - (sigma_t * phi_1) * model_prev_0 - (sigma_t * (phi_1 / h - 1.0)) * D1_0 ) return x_t def multistep_dpm_solver_third_update(self, x, model_prev_list, t_prev_list, t, solver_type="dpmsolver"): """ 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 (1,) t: A pytorch tensor. The ending time, with the shape (1,). solver_type: either 'dpmsolver' or 'taylor'. The type for the high-order solvers. The type slightly impacts the performance. We recommend to use 'dpmsolver' type. Returns: x_t: A pytorch tensor. The approximated solution at time `t`. """ ns = self.noise_schedule 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 = (1.0 / r0) * (model_prev_0 - model_prev_1) D1_1 = (1.0 / r1) * (model_prev_1 - model_prev_2) D1 = D1_0 + (r0 / (r0 + r1)) * (D1_0 - D1_1) D2 = (1.0 / (r0 + r1)) * (D1_0 - D1_1) if self.algorithm_type == "dpmsolver++": phi_1 = torch.expm1(-h) phi_2 = phi_1 / h + 1.0 phi_3 = phi_2 / h - 0.5 x_t = ( (sigma_t / sigma_prev_0) * x - (alpha_t * phi_1) * model_prev_0 + (alpha_t * phi_2) * D1 - (alpha_t * phi_3) * D2 ) else: phi_1 = torch.expm1(h) phi_2 = phi_1 / h - 1.0 phi_3 = phi_2 / h - 0.5 x_t = ( (torch.exp(log_alpha_t - log_alpha_prev_0)) * x - (sigma_t * phi_1) * model_prev_0 - (sigma_t * phi_2) * D1 - (sigma_t * phi_3) * D2 ) return x_t def singlestep_dpm_solver_update( self, x, s, t, order, return_intermediate=False, solver_type="dpmsolver", 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 (1,). t: A pytorch tensor. The ending time, with the shape (1,). 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 'dpmsolver' or 'taylor'. The type for the high-order solvers. The type slightly impacts the performance. We recommend to use 'dpmsolver' 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="dpmsolver"): """ 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 (1,) t: A pytorch tensor. The ending time, with the shape (1,). order: A `int`. The order of DPM-Solver. We only support order == 1 or 2 or 3. solver_type: either 'dpmsolver' or 'taylor'. The type for the high-order solvers. The type slightly impacts the performance. We recommend to use 'dpmsolver' 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="dpmsolver", ): """ 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 'dpmsolver' or 'taylor'. The type for the high-order solvers. The type slightly impacts the performance. We recommend to use 'dpmsolver' 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((1,)).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 def lower_update(x, s, t): return self.dpm_solver_first_update(x, s, t, return_intermediate=True) def higher_update(x, s, t, **kwargs): return self.singlestep_dpm_solver_second_update(x, s, t, r1=r1, solver_type=solver_type, **kwargs) elif order == 3: r1, r2 = 1.0 / 3.0, 2.0 / 3.0 def lower_update(x, s, t): return self.singlestep_dpm_solver_second_update( x, s, t, r1=r1, return_intermediate=True, solver_type=solver_type ) def higher_update(x, s, t, **kwargs): return 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)), ) def norm_fn(v): return torch.sqrt(torch.square(v.reshape((v.shape[0], -1))).mean(dim=-1, keepdim=True)) E = norm_fn((x_higher - x_lower) / delta).max() if torch.all(E <= 1.0): x = x_higher s = t x_prev = x_lower lambda_s = ns.marginal_lambda(s) h = torch.min( theta * h * torch.float_power(E, -1.0 / order).float(), lambda_0 - lambda_s, ) nfe += order print("adaptive solver nfe", nfe) return x def add_noise(self, x, t, noise=None): """ Compute the noised input xt = alpha_t * x + sigma_t * noise. Args: x: A `torch.Tensor` with shape `(batch_size, *shape)`. t: A `torch.Tensor` with shape `(t_size,)`. Returns: xt with shape `(t_size, batch_size, *shape)`. """ alpha_t, sigma_t = self.noise_schedule.marginal_alpha(t), self.noise_schedule.marginal_std(t) if noise is None: noise = torch.randn((t.shape[0], *x.shape), device=x.device) x = x.reshape((-1, *x.shape)) xt = expand_dims(alpha_t, x.dim()) * x + expand_dims(sigma_t, x.dim()) * noise if t.shape[0] == 1: return xt.squeeze(0) else: return xt def inverse( self, x, steps=20, t_start=None, t_end=None, order=2, skip_type="time_uniform", method="multistep", lower_order_final=True, denoise_to_zero=False, solver_type="dpmsolver", atol=0.0078, rtol=0.05, return_intermediate=False, ): """ Inverse the sample `x` from time `t_start` to `t_end` by DPM-Solver. For discrete-time DPMs, we use `t_start=1/N`, where `N` is the total time steps during training. """ t_0 = 1.0 / self.noise_schedule.total_N if t_start is None else t_start t_T = self.noise_schedule.T if t_end is None else t_end assert ( t_0 > 0 and t_T > 0 ), "Time range needs to be greater than 0. For discrete-time DPMs, it needs to be in [1 / N, 1], where N is the length of betas array" return self.sample( x, steps=steps, t_start=t_0, t_end=t_T, order=order, skip_type=skip_type, method=method, lower_order_final=lower_order_final, denoise_to_zero=denoise_to_zero, solver_type=solver_type, atol=atol, rtol=rtol, return_intermediate=return_intermediate, ) def sample( self, x, steps=20, t_start=None, t_end=None, order=2, skip_type="time_uniform", method="multistep", lower_order_final=True, denoise_to_zero=False, solver_type="dpmsolver", atol=0.0078, rtol=0.05, return_intermediate=False, ): """ 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 or DPM-Solver++ ("DPM-Solver-fast" in the paper) with `order = 3`. e.g., DPM-Solver: >>> dpm_solver = DPM_Solver(model_fn, noise_schedule, algorithm_type="dpmsolver") >>> x_sample = dpm_solver.sample(x, steps=steps, t_start=t_start, t_end=t_end, order=3, skip_type='time_uniform', method='singlestep') e.g., DPM-Solver++: >>> dpm_solver = DPM_Solver(model_fn, noise_schedule, algorithm_type="dpmsolver++") >>> 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 `algorithm_type="dpmsolver++"` and `order = 2`. e.g. >>> dpm_solver = DPM_Solver(model_fn, noise_schedule, algorithm_type="dpmsolver++") >>> 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. `dpmsolver` or `taylor`. We recommend `dpmsolver`. 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'. return_intermediate: A `bool`. Whether to save the xt at each step. When set to `True`, method returns a tuple (x0, intermediates); when set to False, method returns only x0. Returns: x_end: A pytorch tensor. The approximated solution at time `t_end`. """ t_0 = 1.0 / self.noise_schedule.total_N if t_end is None else t_end t_T = self.noise_schedule.T if t_start is None else t_start assert ( t_0 > 0 and t_T > 0 ), "Time range needs to be greater than 0. For discrete-time DPMs, it needs to be in [1 / N, 1], where N is the length of betas array" if return_intermediate: assert method in [ "multistep", "singlestep", "singlestep_fixed", ], "Cannot use adaptive solver when saving intermediate values" if self.correcting_xt_fn is not None: assert method in [ "multistep", "singlestep", "singlestep_fixed", ], "Cannot use adaptive solver when correcting_xt_fn is not None" device = x.device intermediates = [] with torch.no_grad(): if method == "adaptive": 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 # Init the initial values. step = 0 t = timesteps[step] t_prev_list = [t] model_prev_list = [self.model_fn(x, t)] if self.correcting_xt_fn is not None: x = self.correcting_xt_fn(x, t, step) if return_intermediate: intermediates.append(x) # Init the first `order` values by lower order multistep DPM-Solver. for step in range(1, order): t = timesteps[step] x = self.multistep_dpm_solver_update( x, model_prev_list, t_prev_list, t, step, solver_type=solver_type, ) if self.correcting_xt_fn is not None: x = self.correcting_xt_fn(x, t, step) if return_intermediate: intermediates.append(x) t_prev_list.append(t) model_prev_list.append(self.model_fn(x, t)) # Compute the remaining values by `order`-th order multistep DPM-Solver. for step in range(order, steps + 1): t = timesteps[step] # We only use lower order for steps < 10 if lower_order_final and steps < 10: step_order = min(order, steps + 1 - step) else: step_order = order x = self.multistep_dpm_solver_update( x, model_prev_list, t_prev_list, t, step_order, solver_type=solver_type, ) if self.correcting_xt_fn is not None: x = self.correcting_xt_fn(x, t, step) if return_intermediate: intermediates.append(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] = t # We do not need to evaluate the final model value. if step < steps: model_prev_list[-1] = self.model_fn(x, 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 step, order in enumerate(orders): s, t = timesteps_outer[step], timesteps_outer[step + 1] timesteps_inner = self.get_time_steps( skip_type=skip_type, t_T=s.item(), t_0=t.item(), N=order, device=device, ) lambda_inner = self.noise_schedule.marginal_lambda(timesteps_inner) 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, s, t, order, solver_type=solver_type, r1=r1, r2=r2) if self.correcting_xt_fn is not None: x = self.correcting_xt_fn(x, t, step) if return_intermediate: intermediates.append(x) else: raise ValueError("Got wrong method {}".format(method)) if denoise_to_zero: t = torch.ones((1,)).to(device) * t_0 x = self.denoise_to_zero_fn(x, t) if self.correcting_xt_fn is not None: x = self.correcting_xt_fn(x, t, step + 1) if return_intermediate: intermediates.append(x) if return_intermediate: return x, intermediates else: 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)]