An AI Approach for Learning the Spectrum of the Laplace-Beltrami Operator

TL;DR

This paper introduces a graph neural network framework that efficiently predicts the Laplace-Beltrami spectrum from CAD meshes, significantly reducing computation time while maintaining accuracy, thus benefiting geometric deep learning and quality control applications.

Contribution

We propose a novel deep learning approach to estimate the LB spectrum from CAD meshes, offering a faster alternative to traditional FEM methods with comparable accuracy.

Findings

Reduces LB spectrum computation time by approximately 5 times.

Achieves competitive accuracy with FEM-based methods.

Provides a curated dataset for training and testing.

Abstract

The spectrum of the Laplace-Beltrami (LB) operator is central in geometric deep learning tasks, capturing intrinsic properties of the shape of the object under consideration. The best established method for its estimation, from a triangulated mesh of the object, is based on the Finite Element Method (FEM), and computes the top k LB eigenvalues with a complexity of O(Nk), where N is the number of points. This can render the FEM method inefficient when repeatedly applied to databases of CAD mechanical parts, or in quality control applications where part metrology is acquired as large meshes and decisions about the quality of each part are needed quickly and frequently. As a solution to this problem, we present a geometric deep learning framework to predict the LB spectrum efficiently given the CAD mesh of a part, achieving significant computational savings without sacrificing accuracy, demonstrating that the LB spectrum is learnable. The proposed Graph Neural Network architecture uses a rich set of part mesh features - including Gaussian curvature, mean curvature, and principal curvatures. In addition to our trained network, we make available, for repeatability, a large curated dataset of real-world mechanical CAD models derived from the publicly available ABC dataset used for training and testing. Experimental results show that our method reduces computation time of the LB spectrum by approximately 5 times over linear FEM while delivering competitive accuracy.