2025/04/22 by Haoqian Li, Li, Haoqian, Jiguang Sun +3 · 1 citation
Computer Science · Mathematics · #35J15 #65N25 #65T50 #68T07 #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2504.15733
openalex publication_date 2025/04/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Eigenvalue problems for elliptic operators play an important role in science and engineering applications, where efficient and accurate numerical computation is essential. In this work, we propose a novel operator inference approach for elliptic eigenvalue problems based on neural network approximations that directly maps computational domains to their associated eigenvalues and eigenfunctions. Motivated by existing neural network architectures and the mathematical characteristics of eigenvalue problems, we represent computational domains as pixelated images and decompose the task into two subtasks: eigenvalue prediction and eigenfunction prediction. For the eigenvalue prediction, we design a convolutional neural network (CNN), while for the eigenfunction prediction, we employ a Fourier Neural Operator (FNO). Additionally, we introduce a critical preprocessing module that integrates domain scaling, detailed boundary pixelization, and main-axis alignment. This preprocessing step not only simplifies the learning task but also enhances the performance of the neural networks. Finally, we present numerical results to demonstrate the effectiveness of the proposed method.