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 promising predictive performance and the associated uncertainty can capture the dynamics of the system appropriately.

翻译：本文提出了一种基于高斯过程的复杂动力学计算机模型（或称模拟器）仿真方法。该方法通过模拟系统在初始（短时）时间步内的数值流形映射来实现——该映射描述了系统从初始状态到下一时刻状态值的演化函数。由此可得到整个流形映射函数的概率分布，其中每个采样值都对应流形映射的近似。基于马尔可夫假设，通过从仿真流形映射（即其后验分布）中抽取样本，并利用该样本从初始条件向前迭代，即可预测模型输出的时间序列。重复该过程并抽取多个样本，即可构建时间序列的分布，其中特定时刻的均值与方差分别对应模型输出预测及其伴随的不确定性。然而，由于高斯过程后验样本具有无限维特性，直接抽取能够表征全域底层函数的样本在计算上不可行。为克服这一限制，可采用随机傅里叶特征生成近似样本。随机傅里叶特征是一种高效逼近核函数并生成高斯过程样本的技术，兼具计算效率与理论保证。将该方法应用于包括经典洛伦兹模型和范德波尔模型在内的多个非线性动力学模拟器的仿真实验，结果表明该方法具有优异的预测性能，且其不确定性估计能有效捕捉系统的动力学特征。