docstr: 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`.
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code: 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
                  + (1.0 / r1)
                  * expand_dims(
                      alpha_t * ((torch.exp(-h) - 1.0) / h + 1.0), 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
                  - (1.0 / r1)
                  * expand_dims(
                      sigma_t * ((torch.exp(h) - 1.0) / h - 1.0), dims
                  )
                  * (model_s1 - model_s)
              )
      if return_intermediate:
          return x_t, {"model_s": model_s, "model_s1": model_s1}
      else:
          return x_t
  
imports: import torch
  import torch.nn.functional
  import math
label_desc: Use image node_id 9 for calling the DPM_Solver obj's underlying member method code with example usage: obj.singlestep_dpm_solver_second_update(x, s, t, r1, model_s, return_intermediate, solver_type) and returns: x_t,  dict, x_t