Graph Convolutional Networks for Learning Laplace-Beltrami Operators

Recovering a high-level representation of geometric data is a fundamental goal in geometric modeling and computer graphics. In this paper, we introduce a data-driven approach to computing the spectrum of the Laplace-Beltrami operator of triangle meshes using graph convolutional networks. Specifically, we train graph convolutional networks on a large-scale dataset of synthetically generated triangle meshes, encoded with geometric data consisting of Voronoi areas, normalized edge lengths, and Gauss map, to infer eigenvalues of 3D shapes. We attempt to address the ability of graph neural networks to capture global shape descriptors–including spectral information–that were previously inaccessible using existing methods from computer vision, and our paper exhibits promising signals suggesting that Laplace-Beltrami eigenvalues on discrete surfaces can be learned. Additionally, we perform ablation studies showing the addition of geometric data leads to improved accuracy.