Operator Inference for Elliptic Eigenvalue Problems

TL;DR

Let $\Omega \subset \mathbb{R}^2$ be a simply connected Lipschitz domain and consider the Dirichlet eigenproblem $-\Delta u = \lambda u$ in $\Omega$, $u=0$ on $\partial\Omega$. The authors propose an operator-learning approach that represents domains as pixelated images and uses a CNN to predict $\lambda$ and a Fourier Neural Operator to predict $u$, following a preprocessing stage of domain scaling to the unit square $[0,1]^2$, main-axis alignment, and edge-aware pixelization. Their experiments on 20,000 domain–eigenpair samples (10,000 smooth, 10,000 non-smooth) show high accuracy for the first eigenpairs and robust generalization to unseen geometries, with dp+ma and higher input resolutions improving performance. This approach enables fast, real-time evaluation of eigenpairs for shape optimization and inverse spectral problems in elliptic PDEs.

Abstract

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.

Figures (13)

• Figure 1: Diagram of dataset generation.
• Figure 1: Structures of the proposed CNNs. Left: $32\times 32$ image. Right: $64\times 64$ image.
• Figure 1: Losses in log scale for the first eigenvalue. Left: $32\times 32$ image. Right: $64\times 64$ image.
• Figure 2: Samples in the dataset. The first 10 eigenvalues and eigenfunctions are shown, as well as their corresponding regions.
• Figure 2: Structure of the proposed FNO.