Improving Gaussian Process Emulators with Boundary Information

Gaussian process (GP) models are widely used as emulators of time-consuming deterministic simulators, which are mostly computer codes that solve partial differential equation (PDE) models of physical systems numerically. In many cases, the functional relationship between the inputs and output of the simulator at parts of the boundary of the experiment domain or input domain can be determined using mathematical analysis, logical reasoning based on physical laws, or a cheap-to-compute low-fidelity simulator, as those subsets of the boundary correspond to simplified physical processes. However, this information is not taken into account in standard stationary GP priors used to construct GP emulators. This chapter considers the problem of constructing a GP emulator that reproduces known input–output relationships of a simulator at some boundary faces of the experiment/input domain, called boundary information/constraints. The proposed boundary modified GP (BMGP) emulator, which employs a nonstationary GP prior with specific forms for the mean and variance functions chosen so that the GP prior satisfies given boundary constraints, is shown to outperform the standard GP emulator based on a stationary GP prior and alternative emulators that satisfy given boundary constraints in two realistic examples.