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

We propose Boundary Ehrenpreis–Palamodov Gaussian Processes (B-EPGPs), a probabilistic framework for constructing Gaussian Processes (GP) priors for linear constant-coefficient Partial Differential Equations (PDE) 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 strictly satisfy both PDE and boundary condition by construction. We provide constructive examples, formal correctness proofs, and experiments showing improved accuracy and reduced runtime/memory compared to existing PDE baselines.