A Gaussian process (GP)-based methodology is proposed to emulate complex dynamical computer models (or simulators). The method relies on emulating the short-time numerical flow map of the system, where the flow map is a function that returns the solution of a dynamical system at a certain time point, given initial conditions. In order to predict the model output times series, a single realisation of the emulated flow map (i.e., its posterior distribution) is taken and used to iterate from the initial condition ahead in time. Repeating this procedure with multiple such draws creates a distribution over the time series whose mean and variance serve as the model output prediction and the associated uncertainty, respectively. However, since there is no known method to draw an exact sample from the GP posterior analytically, we approximate the kernel with random Fourier features and generate approximate sample paths. 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. Additionally, our approach has potential for ``embarrassingly" parallel implementations where one can conduct the iterative predictions performed by a realisation on a single computing node.

翻译：本文提出了一种基于高斯过程的复杂动态计算机模型（或称模拟器）仿真方法。该方法通过模拟系统短时数值流映射（即一个根据初始条件返回动态系统在特定时刻解的映射函数）来实现预测。为获取模型输出的时间序列，我们仅需提取一次模拟流映射的实现（即其后验分布），并基于初始条件向前迭代。重复此过程并多次抽样，即可生成时间序列的分布，其均值与方差分别对应模型输出的预测结果及其不确定性。然而，由于高斯过程后验无法直接获得精确解析样本，我们采用随机傅里叶特征近似核函数以生成近似样本路径。该仿真方法被应用于多个非线性动态模拟器，包括著名的洛伦兹模型和范德波尔模型。结果表明，该方法具有较高的预测性能，且其相关不确定性能够准确捕捉系统动力学特性。此外，本方法具备“令人尴尬”的并行实现潜力——可在单计算节点上独立执行由单个实现生成的迭代预测。