Gaussian Process Regression constrained by Boundary Value Problems

TL;DR

This paper introduces a Gaussian process regression framework constrained by boundary value problems, improving the accuracy and stability of solutions for differential equations with limited scattered data.

Contribution

The authors develop a novel Gaussian process framework that incorporates boundary conditions via spectral kernel expansions, enhancing solution inference for boundary value problems.

Findings

More accurate solution inference compared to unconstrained methods

Enhanced stability in the presence of scattered observations

Reduced computational complexity through low-rank covariance matrices

Abstract

We develop a framework for Gaussian processes regression constrained by boundary value problems. The framework may be applied to infer the solution of a well-posed boundary value problem with a known second-order differential operator and boundary conditions, but for which only scattered observations of the source term are available. Scattered observations of the solution may also be used in the regression. The framework combines co-kriging with the linear transformation of a Gaussian process together with the use of kernels given by spectral expansions in eigenfunctions of the boundary value problem. Thus, it benefits from a reduced-rank property of covariance matrices. We demonstrate that the resulting framework yields more accurate and stable solution inference as compared to physics-informed Gaussian process regression without boundary condition constraints.