Emulating dynamic non-linear simulators using Gaussian processes

TL;DR

This paper presents a method using Gaussian processes to efficiently emulate complex, non-linear dynamic systems over time, enabling faster analysis of models like climate or brain simulations with quantified uncertainty.

Contribution

The paper introduces a novel approach to emulate non-linear time series systems using Gaussian processes, accounting for input uncertainty and correlations over time.

Findings

High predictive accuracy on Lorenz and Van der Pol systems

Uncertainty measures reflect system predictability

Method reduces computational cost of complex simulations

Abstract

The dynamic emulation of non-linear deterministic computer codes where the output is a time series, possibly multivariate, is examined. Such computer models simulate the evolution of some real-world phenomenon over time, for example models of the climate or the functioning of the human brain. The models we are interested in are highly non-linear and exhibit tipping points, bifurcations and chaotic behaviour. However, each simulation run could be too time-consuming to perform analyses that require many runs, including quantifying the variation in model output with respect to changes in the inputs. Therefore, Gaussian process emulators are used to approximate the output of the code. To do this, the flow map of the system under study is emulated over a short time period. Then, it is used in an iterative way to predict the whole time series. A number of ways are proposed to take into account the uncertainty of inputs to the emulators, after fixed initial conditions, and the correlation between them through the time series. The methodology is illustrated with two examples: the highly non-linear dynamical systems described by the Lorenz and Van der Pol equations. In both cases, the predictive performance is relatively high and the measure of uncertainty provided by the method reflects the extent of predictability in each system.