Emulating complex dynamical simulators with random Fourier features

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.