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.mode