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

TMLR Paper7479 Authors

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 probabilistic framework for constructing GP priors for linear constant-coefficient PDE systems with linear boundary conditions that can be conditioned on a finite data set. Starting from the Ehrenpreis--Palamodov representation, we learn the free parameters from data and enforce boundary conditions analytically for piecewise-flat boundaries. This yields priors (and posteriors) whose sample paths satisfy the PDE and the enforced boundary conditions pointwise by construction. We provide constructive examples, formal correctness proofs, and experiments showing improved accuracy and reduced runtime/memory compared to existing GP-PDE baselines.

Submission Type: Regular submission (no more than 12 pages of main content)

Assigned Action Editor: ~Yi_Zhou2

Submission Number: 7479