Gaussian Processes for Design Using Nondifferentiable Functions With Jump Discontinuities

Abstract

Available Gaussian processes (GPs) are inappropriate when used to globally emulate and optimize nondifferentiable functions with jump discontinuities. This is a substantial limitation as many contemporary engineering challenges are underpinned by functions with these characteristics and would benefit from a data-efficient emulator (e.g., contact problems in mechanics and material properties because of phase changes). Available GP models are inappropriate as jump discontinuities and nondifferentiabilities are local phenomena that cannot be modeled through covariance structures that describe a function’s global behavior. To overcome this limitation, we introduce the discontinuous Gaussian process (DCGP) model that involves learning a parametric, yet flexible function on the inputs to ensure that the residuals of the random process are continuous and differentiable. Through validation on a set of six test problems and one engineering problem, we show that the DCGP model outperforms available models that use multiple local emulators. The performance of the DCGP model is measured in terms of predictive fidelity and optimization efficiency. In addition, we show that there exist physical systems for which the correlation between its outputs can be modeled across discontinuities and nondifferentiabilities (i.e., globally). We believe this to be a profound insight as it suggests that statistical models, such as DCGP, could be used to expedite the discovery of emerging properties even for discontinuous and nondifferentiable functions.