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

Created by
  • Haebom

Author

Yulin An, Enrique del Castillo

Outline

In this paper, we present an efficient geometric deep learning framework for estimating the spectrum of the Laplace-Beltrami (LB) operator, which captures the intrinsic properties of objects in geometric deep learning. The conventional finite element method (FEM) has a computational complexity of O(Nk), which is inefficient for processing large-scale mesh data. In this study, we utilize rich mesh features such as Gaussian curvature, mean curvature, and principal curvature using graph neural networks (GNNs) to estimate the LB spectrum. Experimental results show that the proposed method is about 5 times faster than FEM and provides competitive accuracy. In addition, we also release a large-scale real mechanical CAD model dataset built on the ABC dataset used for training and testing for reproducibility.

Takeaways, Limitations

Takeaways:
It provides LB spectrum calculation speed 5 times faster than the existing FEM method.
We present an efficient LB spectral prediction framework based on GNN.
We increased the reproducibility of our research by releasing real machine CAD model datasets.
Effective for applications requiring large-scale mesh data processing (e.g. CAD parts databases, quality control).
Limitations:
The accuracy of the proposed method may not be exactly the same as that of FEM (although it is stated that it provides competitive accuracy, it is not specified exactly how much difference it makes).
Additional validation of the generalization performance of the dataset used may be required.
It may only be effective on certain types of meshes.
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