Emulating computationally expensive dynamical simulators using Gaussian processes

A Gaussian process (GP)-based methodology is proposed to emulate computationally expensive dynamical computer models or simulators. The method relies on emulating the short-time numerical flow map of the model. The flow map returns the solution of a dynamic system at an arbitrary time for a given initial condition. The prediction of the flow map is performed via a GP whose kernel is estimated using random Fourier features. This gives a distribution over the flow map such that each realisation serves as an approximation to the flow map. A realisation is then employed in an iterative manner to perform one-step ahead predictions and forecast the whole time series. Repeating this procedure with multiple draws from the emulated flow map provides a probability distribution over the time series. The mean and variance of that distribution are used as the model output prediction and a measure of the associated uncertainty, respectively. The proposed method is used to emulate several dynamic non-linear simulators including the well-known Lorenz attractor and van der Pol oscillator. The results show that our approach has a high prediction performance in emulating such systems with an accurate representation of the prediction uncertainty.