Gaussian Process Priors for Boundary Value Problems of Linear Partial Differential Equations

TL;DR

This paper introduces Boundary Ehrenpreis--Palamodov Gaussian Processes (B-EPGPs), a new probabilistic framework for solving linear PDE systems with boundary conditions, showing improved accuracy and efficiency over existing methods.

Contribution

It presents a novel GP prior construction for linear PDEs with boundary conditions, with formal proofs and empirical validation.

Findings

Significant accuracy improvements over state-of-the-art methods

Reduced computational resources required

Effective handling of boundary conditions in PDE systems

Abstract

Working with systems of partial differential equations (PDEs) is a fundamental task in computational science. Well-posed systems are addressed by numerical solvers or neural operators, whereas systems described by data are often addressed by PINNs or Gaussian processes. In this work, we propose Boundary Ehrenpreis--Palamodov Gaussian Processes (B-EPGPs), a novel probabilistic framework for constructing GP priors that satisfy both general systems of linear PDEs with constant coefficients and linear boundary conditions and can be conditioned on a finite data set. 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 and computational resource improvements over state-of-the-art approaches.