Gaussian Processes for Non-Differentiable Functions With Jump Discontinuities
Anton v. Beek · 2024
Abstract Gaussian processes (GPs) are used in many disciplines, but the approximation of functions with jump discontinuities and non-differentiabilities remains a challenge. This is a substantial limitation as many contemporary scientific and engineering challenges involve functions with these characteristics (e.g., contact problems in mechanics, and material properties because of phase changes). Available GP models are inappropriate as jump discontinuities and non-differentiabilities 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 training data 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 has superior performance over available models and that there exist physical systems for which the correlation between a function’s outputs can be modeled across discontinuities and non-differentiabilities (i.e., globally). We believe the latter to be a profound insight as it suggests that statistical models, such as DCGP, could be used to expedite scientific discovery even for discontinuous and non-differentiable functions.