用图神经网络预测拉普拉斯-贝尔特拉米谱，提速5倍且精度不降。
An AI Approach for Learning the Spectrum of the Laplace-Beltrami Operator
- 构建GNN模型，输入几何曲率等网格特征，直接预测LB谱
- 相比线性FEM，计算速度提升约5倍，误差可控
- 适合需要快速批量处理机械零件的质检场景
拉普拉斯-贝尔特拉米（LB）算子的谱在几何深度学习中至关重要，能捕捉物体形状的内在特性。现有主流方法基于有限元法（FEM），从三角网格估计前k个LB特征值，复杂度为O(Nk)，其中N为点数。该方法在重复处理大量机械零件CAD模型或需快速决策的零件检测场景下效率低下。为此，本文提出一种几何深度学习框架，通过训练图神经网络，仅需输入零件网格即可高效预测其LB谱，实现显著计算加速且保持精度。所提架构融合高斯曲率、平均曲率及主曲率等丰富几何特征。此外，我们公开了一个由公开ABC数据集构建的大规模真实机械零件CAD模型数据集，以保证可复现性。实验表明，本方法相较线性FEM将计算时间缩短约5倍，同时保持竞争力的准确性。
原文摘要 · Abstract (English)
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.
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