Physics-informed Gaussian Process Regression in Solving Eigenvalue Problem of Linear Operators

TL;DR

This paper introduces a physics-informed Gaussian Process Regression method to identify eigenvalues and eigenfunctions of linear operators, overcoming challenges with trivial predictions in eigenvalue problems.

Contribution

It develops a novel transfer function indicator based on the GP posterior that detects eigenvalues by covariance structure, enabling eigenproblem solutions.

Findings

The covariance is non-trivial only at true eigenvalues.

Samples from the posterior lie in the eigenspace of the operator.

The method effectively solves linear and nonlinear eigenvalue problems.

Abstract

Applying Physics-Informed Gaussian Process Regression to the eigenvalue problem $(\mathcal{L}-\lambda)u = 0$ poses a fundamental challenge, where the null source term results in a trivial predictive mean and a degenerate marginal likelihood. Drawing inspiration from system identification, we construct a transfer function-type indicator for the unknown eigenvalue/eigenfunction using the physics-informed Gaussian Process posterior. We demonstrate that the posterior covariance is only non-trivial when $\lambda$ corresponds to an eigenvalue of the partial differential operator $\mathcal{L}$, reflecting the existence of a non-trivial eigenspace, and any sample from the posterior lies in the eigenspace of the linear operator. We demonstrate the effectiveness of the proposed approach through several numerical examples with both linear and non-linear eigenvalue problems.