Gaussian Process Priors for Systems of Linear Partial Differential Equations with Constant Coefficients

TL;DR

This paper introduces a novel Gaussian process prior, EPGP, for solving linear PDE systems with constant coefficients, enabling efficient and accurate inference of solutions from data, including noisy and boundary conditions.

Contribution

The paper develops a new family of Gaussian process priors based on the Ehrenpreis-Palamodov principle, providing exact solutions for linear PDE systems and scalable algorithms for large datasets.

Findings

EPGP outperforms existing methods in speed and accuracy.

Sparse EPGP effectively learns spectral frequencies for large data.

Demonstrated on heat, wave, and Maxwell's equations.

Abstract

Partial differential equations (PDEs) are important tools to model physical systems and including them into machine learning models is an important way of incorporating physical knowledge. Given any system of linear PDEs with constant coefficients, we propose a family of Gaussian process (GP) priors, which we call EPGP, such that all realizations are exact solutions of this system. We apply the Ehrenpreis-Palamodov fundamental principle, which works as a non-linear Fourier transform, to construct GP kernels mirroring standard spectral methods for GPs. Our approach can infer probable solutions of linear PDE systems from any data such as noisy measurements, or pointwise defined initial and boundary conditions. Constructing EPGP-priors is algorithmic, generally applicable, and comes with a sparse version (S-EPGP) that learns the relevant spectral frequencies and works better for big data sets. We demonstrate our approach on three families of systems of PDEs, the heat equation, wave equation, and Maxwell's equations, where we improve upon the state of the art in computation time and precision, in some experiments by several orders of magnitude.