Gaussian Process Regression constrained by Boundary Value Problems
This work addresses the problem of accurately and stably inferring solutions to boundary value problems for researchers and engineers working with differential equations and sparse data.
This paper introduces a Gaussian process regression framework for inferring solutions to boundary value problems when only scattered observations of the source term or solution are available. It achieves more accurate and stable solution inference compared to physics-informed Gaussian process regression that lacks boundary condition constraints.
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.