Gaussian Process Priors for Boundary Value Problems of Linear Partial Differential Equations (2411.16663v1)

Abstract: Solving systems of partial differential equations (PDEs) is a fundamental task in computational science, traditionally addressed by numerical solvers. Recent advancements have introduced neural operators and physics-informed neural networks (PINNs) to tackle PDEs, achieving reduced computational costs at the expense of solution quality and accuracy. Gaussian processes (GPs) have also been applied to linear PDEs, with the advantage of always yielding precise solutions. In this work, we propose Boundary Ehrenpreis-Palamodov Gaussian Processes (B-EPGPs), a novel framework for constructing GP priors that satisfy both general systems of linear PDEs with constant coefficients and linear boundary conditions. We explicitly construct GP priors for representative PDE systems with practical boundary conditions. Formal proofs of correctness are provided and empirical results demonstrating significant accuracy improvements over state-of-the-art neural operator approaches.