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

Jianlei Huang, Marc Harkonen, Markus Lange-Hegermann, Bogdan Raita

Keywords: Gaussian Processes, Partial Differential Equations, Regression

TL;DR: GP priors for general systems of linear PDEs with constant coefficients and linear boundary conditions.

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. The Ehrenpreis--Palamodov theorem provides the functional form of the solution, but leaves parameters of this functional form unknown. An optimization finds these parameters and we provide an algorithm to modify this function form to satisfy 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 and computational resource improvements over state-of-the-art approaches.

Primary Area: probabilistic methods (Bayesian methods, variational inference, sampling, UQ, etc.)

Submission Number: 12002