Operator Inference for Elliptic Eigenvalue Problems

Abstract

Efficient computation of elliptic eigenvalue problems is critical in science and engineering. In this paper, we propose a novel operator learning framework that directly maps arbitrary domains to their associated eigenvalues and eigenfunctions. Our approach employs a hybrid architecture combining Convolutional Neural Networks (CNNs) and Fourier Neural Operators (FNOs), enhanced by a robust preprocessing pipeline—including main axis alignment and detailed pixelization, that normalizes inputs to significantly simplify the learning task. Furthermore, we introduce a geometric-aware loss function specifically designed to domain variations and the inherent sign ambiguity of eigenfunctions, ensuring stable and consistent training. Extensive numerical experiments demonstrate that our method achieves superior accuracy and computational efficiency compared to traditional numerical solvers, particularly in many-query scenarios.