Gaussian Process Priors for Boundary Value Problems of Linear Partial Differential Equations
This work addresses a fundamental challenge in computational science for researchers and practitioners dealing with PDE systems, offering a novel probabilistic approach that is not incremental but introduces a new framework.
The authors tackled the problem of solving linear partial differential equations (PDEs) with boundary conditions by proposing Boundary Ehrenpreis--Palamodov Gaussian Processes (B-EPGPs), a probabilistic framework that constructs Gaussian process priors satisfying both PDEs and boundary conditions, resulting in significant accuracy and computational resource improvements over state-of-the-art methods.
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.