A Physics-Informed Graph Neural Network for Computing Laplace–Beltrami Eigenfunctions on Manifolds

Eigenfunctions and the spectrum of the Laplace-Beltrami operator are fundamental tools in geometry processing, providing a robust framework for analyzing complex geometries. This paper introduces an incremental framework specifically designed for the computation of the leading eigenpairs of the Laplace-Beltrami operator of 2-manifolds represented by triangular surface meshes in R-3. The proposed approach relies on a discrete physics-informed neural network (dPINN), which leverages Graph Neural Networks to effectively model the intrinsic geometry and topology of the manifold. dPINNs are ideally suited for learning on a triangular surface mesh in R-3 because they leverage its discrete representation and numerical differentiation. Conversely, continuous PINNs (cPINNs) prove inadequate when the spatial domain (manifold) is not readily available for arbitrary sampling. Numerical experiments on several 2-manifolds in R-2/R-3 demonstrate the accuracy, robustness to geometric and topological variations, and generalization capabilities of the proposed neural network approach in case of 2-manifolds with parametric representations. The presented method aims to demonstrate how learning-based approaches can improve upon traditional numerical methods under adverse meshing conditions or domain discretization changes.