Deep Eigenspace Network for Parametric Non-self-adjoint Eigenvalue Problems

TL;DR

This paper introduces a Deep Eigenspace Network (DEN) that efficiently learns eigenspaces for parametric non-self-adjoint eigenvalue problems, addressing spectral instability and mode switching.

Contribution

The paper proposes a novel DEN architecture combining Fourier Neural Operators, adaptive POD bases, and cross-mode mixing to improve spectral problem solutions.

Findings

DEN effectively captures complex spectral dependencies.

The method demonstrates high efficiency and accuracy in numerical experiments.

Lipschitz continuity of eigenspaces with respect to parameters is established.

Abstract

We consider operator learning for efficiently solving parametric non-self-adjoint eigenvalue problems. To overcome the spectral instability and mode switching associated with non-self-adjoint operators, we choose to learn the eigenspace rather than individual eigenfunctions. In particular, we propose a Deep Eigenspace Network (DEN) architecture integrating Fourier Neural Operators, geometry-adaptive POD bases, and explicit banded cross-mode mixing mechanism to capture complex spectral dependencies. We apply DEN to the non-self-adjoint Steklov eigenvalue problem and prove the Lipschitz continuity of the eigenspace with respect to the parameter. Furthermore, we derive error bounds for the eigenvalues. Numerical experiments validate that DEN is highly effective and efficient.