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

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.