Data-driven reduced order modeling for parametric PDE eigenvalue problems using Gaussian process regression

TL;DR

This paper introduces a data-driven reduced basis method utilizing Gaussian process regression for efficient approximation of parametric eigenvalue problems in PDEs, enabling rapid online predictions after offline training.

Contribution

It combines POD-based reduced basis construction with GPR for eigenvalue and eigenvector coefficient approximation, applicable to affine and non-affine problems.

Findings

Robustness demonstrated on various eigenvalue problems.

Efficient online predictions with trained GPR models.

Applicable to both affine and non-affine parameter dependencies.

Abstract

In this article, we propose a data-driven reduced basis (RB) method for the approximation of parametric eigenvalue problems. The method is based on the offline and online paradigms. In the offline stage, we generate snapshots and construct the basis of the reduced space, using a POD approach. Gaussian process regressions (GPR) are used for approximating the eigenvalues and projection coefficients of the eigenvectors in the reduced space. All the GPR corresponding to the eigenvalues and projection coefficients are trained in the offline stage, using the data generated in the offline stage. The output corresponding to new parameters can be obtained in the online stage using the trained GPR. The proposed algorithm is used to solve affine and non-affine parameter-dependent eigenvalue problems. The numerical results demonstrate the robustness of the proposed non-intrusive method.