Emulating computer models with step-discontinuous outputs using Gaussian processes

TL;DR

This paper explores how Gaussian processes can be adapted to emulate functions with discontinuities, such as bifurcations, by developing specialized kernels and input warping techniques, improving their ability to model abrupt changes.

Contribution

The paper introduces novel methods including adapted kernels and input warping to enhance Gaussian process emulation of step-discontinuous functions.

Findings

Specialized kernels improve discontinuity modeling.

Input warping enhances GP performance on sharp jumps.

Proposed methods outperform standard GPs in examples.

Abstract

In many real-world applications we are interested in approximating costly functions that are analytically unknown, e.g. complex computer codes. An emulator provides a fast approximation of such functions relying on a limited number of evaluations. Gaussian processes (GPs) are commonplace emulators due to their statistical properties such as the ability to estimate their own uncertainty. GPs are essentially developed to fit smooth, continuous functions. However, the assumptions of continuity and smoothness is unwarranted in many situations. For example, in computer models where bifurcations or tipping points occur, the outputs can be discontinuous. This work examines the capacity of GPs for emulating step-discontinuous functions. Several approaches are proposed for this purpose. Two special covariance functions/kernels are adapted with the ability to model discontinuities. They are the neural network and Gibbs kernels whose properties are demonstrated using several examples. Another approach, which is called warping, is to transform the input space into a new space where a GP with a standard kernel, such as the Matern family, is able to predict the function well. The transformation is perform by a parametric map whose parameters are estimated by maximum likelihood. The results show that the proposed approaches have superior performance to GPs with standard kernels in capturing sharp jumps in the true function.