Emulating dynamic non-linear simulators using Gaussian processes.

In this paper, we examine the emulation of non-linear deterministic computercodes where the output is a time series, possibly multivariate. Such computermodels simulate the evolution of some real-world phenomena over time, forexample models of the climate or the functioning of the human brain. The modelswe are interested in are highly non-linear and exhibit tipping points,bifurcations and chaotic behaviour. Each simulation run is too time-consumingto perform naive uncertainty quantification. We therefore build emulators usingGaussian processes to model the output of the code. We use the Gaussian processto predict one-step ahead in an iterative way over the whole time series. Weconsider a number of ways to propagate uncertainty through the time seriesincluding both the uncertainty of inputs to the emulators at time t and thecorrelation between them. The methodology is illustrated with a number ofexamples. These include the highly non-linear dynamical systems described bythe Lorenz and Van der Pol equations. In both cases we will show that we notonly have very good predictive performance but also have measures ofuncertainty that reflect what is known about predictability in each system.