A Gaussian process (GP)-based methodology is proposed to emulate complex dynamical computer models (or simulators). The method relies on emulating the numerical flow map of the system over an initial (short) time step, where the flow map is a function that describes the evolution of the system from an initial condition to a subsequent value at the next time step. This yields a probabilistic distribution over the entire flow map function, with each draw offering an approximation to the flow map. The model output times series is then predicted (under the Markov assumption) by drawing a sample from the emulated flow map (i.e., its posterior distribution) and using it to iterate from the initial condition ahead in time. Repeating this procedure with multiple such draws creates a distribution over the time series. The mean and variance of this distribution at a specific time point serve as the model output prediction and the associated uncertainty, respectively. However, drawing a GP posterior sample that represents the underlying function across its entire domain is computationally infeasible, given the infinite-dimensional nature of this object. To overcome this limitation, one can generate such a sample in an approximate manner using random Fourier features (RFF). RFF is an efficient technique for approximating the kernel and generating GP samples, offering both computational efficiency and theoretical guarantees. The proposed method is applied to emulate several dynamic nonlinear simulators including the well-known Lorenz and van der Pol models. The results suggest that our approach has a high predictive performance and the associated uncertainty can capture the dynamics of the system accurately.

翻译：提出了一种基于高斯过程（GP）的方法来模拟复杂动力学计算机模型（或仿真器）。该方法依赖于在初始（短）时间步长上模拟系统的数值流映射，其中流映射是一个函数，描述系统从初始状态到下一个时间步长后续值的演变过程。这为整个流映射函数生成了一个概率分布，每次抽取都提供了对流映射的近似。然后，在马尔可夫假设下，通过从模拟的流映射（即其后验分布）中抽取一个样本，并利用该样本从初始条件向前迭代，从而预测模型输出时间序列。通过多次重复这一过程，可以生成时间序列上的分布。该分布在特定时间点的均值和方差分别作为模型输出的预测值和相关不确定性。然而，由于GP后验样本的无限维特性，在其整个定义域内抽取代表底层函数的样本在计算上是不可行的。为克服这一限制，可以采用随机傅里叶特征（RFF）以近似方式生成这样的样本。RFF是一种高效近似核函数并生成GP样本的技术，兼具计算效率和理论保证。将该方法应用于多个动态非线性仿真器的模拟，包括著名的洛伦兹模型和范德波尔模型。结果表明，我们的方法具有较高的预测性能，且相关不确定性能够准确捕捉系统的动态特性。