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

H. Li, J. Sun, Z. Zhang

TL;DR

This work tackles parametric non-selfadjoint eigenvalue problems by shifting focus from unstable eigenfunctions to the stable invariant eigensubspace. The Deep Eigenspace Network (DEN) combines Fourier neural operators with geometry-adaptive POD bases and a banded low-rank cross-mode mixer to capture inter-modal spectral couplings, enabling robust subspace prediction and subsequent Rayleigh-Ritz reconstruction of eigenpairs. The authors prove Lipschitz continuity of the eigensubspace with respect to parameters and derive eigenvalue reconstruction error bounds, and demonstrate strong zero-shot generalization across meshes and robustness to varying wavenumbers via subspace embedding. The approach offers a fast, discretization-agnostic surrogate for parametric Steklov problems with potential applications in MOR, GFEM enrichment, and inverse-scattering workflows.

Abstract

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.

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

TL;DR

This work tackles parametric non-selfadjoint eigenvalue problems by shifting focus from unstable eigenfunctions to the stable invariant eigensubspace. The Deep Eigenspace Network (DEN) combines Fourier neural operators with geometry-adaptive POD bases and a banded low-rank cross-mode mixer to capture inter-modal spectral couplings, enabling robust subspace prediction and subsequent Rayleigh-Ritz reconstruction of eigenpairs. The authors prove Lipschitz continuity of the eigensubspace with respect to parameters and derive eigenvalue reconstruction error bounds, and demonstrate strong zero-shot generalization across meshes and robustness to varying wavenumbers via subspace embedding. The approach offers a fast, discretization-agnostic surrogate for parametric Steklov problems with potential applications in MOR, GFEM enrichment, and inverse-scattering workflows.

Abstract

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.
Paper Structure (37 sections, 8 theorems, 71 equations, 8 figures, 5 tables)

This paper contains 37 sections, 8 theorems, 71 equations, 8 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Deep Eigenspace Network
  3. Structure of the Fourier Neural Operator
  4. Modeling Spectral Couplings via Cross-Mode Mixing
  5. Geometry-Adaptive Spectral Learning
  6. Optimality of the POD Basis
  7. Enhancing Expressivity via Cross-Mode Mixing
  8. On the Basis Selection
  9. Banded Low-Rank Parameterization
  10. Low-Rank Property of the Spectral Operator
  11. Optimization Stability via Banded Constraints
  12. Mathematical Formulation
  13. Methodology
  14. Problem Setup
  15. The Non-Selfadjoint Steklov Eigenvalue Problem
  16. ...and 22 more sections

Key Result

Proposition 2.1

Let $A_{\Psi} \in \mathbb{C}^{K \times K}$ be the spectral representation of a target operator. Let $\mathcal{D} \subset \mathbb{C}^{K \times K}$ be the subspace of diagonal matrices, and let $\mathcal{M} = \mathbb{C}^{K \times K}$ be the space of arbitrary matrices. Define the approximation errors where $\| \cdot \|_F$ denotes the Frobenius norm. If $A_{\Psi}$ possesses non-zero off-diagonal ent

Figures (8)

  • Figure 1: Structure of Deep Eigenspace Network.
  • Figure 1: Samples of generated $n(x)$
  • Figure 1: Real part of eigenfunctions (top), their projection onto the predicted eigensubspace (middle) and relative errors (bottom).
  • Figure 2: Imaginary part of eigenfunctions (top), their projection onto the predicted eigensubspace (middle) and relative errors (bottom).
  • Figure 4: Interpolation error of eigenvalues and eigenfunctions
  • ...and 3 more figures

Theorems & Definitions (18)

  • Proposition 2.1: Irreducible Error of Diagonal Approximations
  • Proof 1
  • Definition 2.2: Entangled Basis
  • Theorem 2.3: Effective Rank Contraction
  • Proof 2
  • Theorem 2.4: Optimality of the Output-Aligned Basis
  • Proof 3
  • Lemma 4.4: Resolvent Stability
  • Proof 4
  • Remark 4.5: Practical Implications
  • ...and 8 more