Gaussian Process Regression for Inverse Problems in Linear PDEs

TL;DR

This paper presents a novel Gaussian process-based method for efficiently solving inverse problems governed by linear PDEs, demonstrated through wave speed identification with high accuracy.

Contribution

It introduces a new algorithm combining Gaussian processes with algebraic priors for inverse PDE problems, implemented via Macaulay2 software.

Findings

High accuracy in wave speed identification

Enhanced computational efficiency

Effective handling of noisy data

Abstract

This paper introduces a computationally efficient algorithm in system theory for solving inverse problems governed by linear partial differential equations (PDEs). We model solutions of linear PDEs using Gaussian processes with priors defined based on advanced commutative algebra and algebraic analysis. The implementation of these priors is algorithmic and achieved using the Macaulay2 computer algebra software. An example application includes identifying the wave speed from noisy data for classical wave equations, which are widely used in physics. The method achieves high accuracy while enhancing computational efficiency.