Deep Eigenspace Network and Its Application to Parametric Non-selfadjoint Eigenvalue Problems

We consider operator learning for efficiently solving parametric non-selfadjoint eigenvalue problems. To overcome the spectral instability and mode switching inherent in non-selfadjoint operators, we introduce a hybrid framework that learns the stable invariant eigensubspace mapping rather than individual eigenfunctions. We proposed a Deep Eigenspace Network (DEN) architecture integrating Fourier Neural Operators, geometry-adaptive POD bases, and explicit banded cross-mode mixing mechanisms to capture complex spectral dependencies on unstructured meshes. We apply DEN to the parametric non-selfadjoint Steklov eigenvalue problem and provide theoretical proofs for the Lipschitz continuity of the eigensubspace with respect to the parameters. In addition, we derive error bounds for the reconstruction of the eigenspace. Numerical experiments validate DEN's high accuracy and zero-shot generalization capabilities across different discretizations.