Gaussian Process Priors for Boundary Value Problems of Linear Partial Differential Equations
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.
Keywords
Cite
@article{arxiv.2411.16663,
title = {Gaussian Process Priors for Boundary Value Problems of Linear Partial Differential Equations},
author = {Jianlei Huang and Marc Härkönen and Markus Lange-Hegermann and Bogdan Raiţă},
journal= {arXiv preprint arXiv:2411.16663},
year = {2025}
}
Comments
36 pages, 18 figures. Code available at $\href{https://github.com/Jimmy000207/Boundary-EPGP}{\text{this https URL}}$. The paper and all ancillary files are released under CC-BY