Linearly Constrained Gaussian Processes with Boundary Conditions

TL;DR

This paper introduces a method to construct multi-output Gaussian process priors that exactly satisfy linear PDE systems and boundary conditions, using an algorithmic approach with Gr"obner bases, without approximations.

Contribution

It presents a novel, exact construction of Gaussian process priors constrained by linear PDEs and boundary conditions, leveraging algebraic methods for the first time.

Findings

Constructs Gaussian process priors that exactly satisfy PDEs and boundary conditions.

Uses Gr"obner bases for an exact, algorithmic construction without approximations.

Enables efficient Bayesian modeling of systems governed by linear PDEs.

Abstract

One goal in Bayesian machine learning is to encode prior knowledge into prior distributions, to model data efficiently. We consider prior knowledge from systems of linear partial differential equations together with their boundary conditions. We construct multi-output Gaussian process priors with realizations in the solution set of such systems, in particular only such solutions can be represented by Gaussian process regression. The construction is fully algorithmic via Gr\"obner bases and it does not employ any approximation. It builds these priors combining two parametrizations via a pullback: the first parametrizes the solutions for the system of differential equations and the second parametrizes all functions adhering to the boundary conditions.