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

• URL: http://arxiv.org/abs/2507.07073v1
• Date: Wed, 09 Jul 2025 17:31:18 GMT
• Title: An AI Approach for Learning the Spectrum of the Laplace-Beltrami Operator
• Authors: Yulin An, Enrique del Castillo,
• Abstract summary: We present a deep learning framework to predict the Laplace-Beltrami (LB) spectrum efficiently given the CAD mesh of a part.<n>The proposed Graph Neural Network architecture uses a rich set of part mesh features - including Gaussian curvature, mean curvature, and principal curvatures.<n> Experimental results show that our method reduces computation time of the LB spectrum by approximately 5 times over linear FEM.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

Related papers

• Geometric Operator Learning with Optimal Transport [77.16909146519227]
  We propose integrating optimal transport (OT) into operator learning for partial differential equations (PDEs) on complex geometries.<n>For 3D simulations focused on surfaces, our OT-based neural operator embeds the surface geometry into a 2D parameterized latent space.<n> Experiments with Reynolds-averaged Navier-Stokes equations (RANS) on the ShapeNet-Car and DrivAerNet-Car datasets show that our method achieves better accuracy and also reduces computational expenses.
  arXiv  Detail & Related papers  (2025-07-26T21:28:25Z)
• Performance Analysis of Convolutional Neural Network By Applying Unconstrained Binary Quadratic Programming [0.0]
  Convolutional Neural Networks (CNNs) are pivotal in computer vision and Big Data analytics but demand significant computational resources when trained on large-scale datasets.<n>We propose a hybrid optimization method that combines Unconstrained Binary Quadratic Programming (UBQP) with Gradient Descent (SGD) to accelerate CNN training.<n>Our approach achieves a 10--15% accuracy improvement over a standard BP-CNN baseline while maintaining similar execution times.
  arXiv  Detail & Related papers  (2025-05-30T21:25:31Z)
• DimOL: Dimensional Awareness as A New 'Dimension' in Operator Learning [60.58067866537143]
  We introduce DimOL (Dimension-aware Operator Learning), drawing insights from dimensional analysis.<n>To implement DimOL, we propose the ProdLayer, which can be seamlessly integrated into FNO-based and Transformer-based PDE solvers.<n> Empirically, DimOL models achieve up to 48% performance gain within the PDE datasets.
  arXiv  Detail & Related papers  (2024-10-08T10:48:50Z)
• Iterative Sizing Field Prediction for Adaptive Mesh Generation From Expert Demonstrations [49.173541207550485]
  Adaptive Meshing By Expert Reconstruction (AMBER) is an imitation learning problem. AMBER combines a graph neural network with an online data acquisition scheme to predict the projected sizing field of an expert mesh. We experimentally validate AMBER on 2D meshes and 3D meshes provided by a human expert, closely matching the provided demonstrations and outperforming a single-step CNN baseline.
  arXiv  Detail & Related papers  (2024-06-20T10:01:22Z)
• Large-Scale OD Matrix Estimation with A Deep Learning Method [70.78575952309023]
  The proposed method integrates deep learning and numerical optimization algorithms to infer matrix structure and guide numerical optimization. We conducted tests to demonstrate the good generalization performance of our method on a large-scale synthetic dataset.
  arXiv  Detail & Related papers  (2023-10-09T14:30:06Z)
• Robust Visual Odometry Using Position-Aware Flow and Geometric Bundle Adjustment [16.04240592057438]
  A novel optical flow network (PANet) built on a position-aware mechanism is proposed first. Then, a novel system that jointly estimates depth, optical flow, and ego-motion without a typical network to learning ego-motion is proposed. Experiments show that the proposed system not only outperforms other state-of-the-art methods in terms of depth, flow, and VO estimation.
  arXiv  Detail & Related papers  (2021-11-22T12:05:27Z)
• FG-Net: Fast Large-Scale LiDAR Point CloudsUnderstanding Network Leveraging CorrelatedFeature Mining and Geometric-Aware Modelling [15.059508985699575]
  FG-Net is a general deep learning framework for large-scale point clouds understanding without voxelizations. We propose a deep convolutional neural network leveraging correlated feature mining and deformable convolution based geometric-aware modelling. Our approaches outperform state-of-the-art approaches in terms of accuracy and efficiency.
  arXiv  Detail & Related papers  (2020-12-17T08:20:09Z)
• Wide-band butterfly network: stable and efficient inversion via multi-frequency neural networks [1.2891210250935143]
  We introduce an end-to-end deep learning architecture called the wide-band butterfly network (WideBNet) for approximating the inverse scattering map from wide-band scattering data. This architecture incorporates tools from computational harmonic analysis, such as the butterfly factorization, and traditional multi-scale methods, such as the Cooley-Tukey FFT algorithm.
  arXiv  Detail & Related papers  (2020-11-24T21:48:43Z)
• A Point-Cloud Deep Learning Framework for Prediction of Fluid Flow Fields on Irregular Geometries [62.28265459308354]
  Network learns end-to-end mapping between spatial positions and CFD quantities. Incompress laminar steady flow past a cylinder with various shapes for its cross section is considered. Network predicts the flow fields hundreds of times faster than our conventional CFD.
  arXiv  Detail & Related papers  (2020-10-15T12:15:02Z)
• Neural Subdivision [58.97214948753937]
  This paper introduces Neural Subdivision, a novel framework for data-driven coarseto-fine geometry modeling. We optimize for the same set of network weights across all local mesh patches, thus providing an architecture that is not constrained to a specific input mesh, fixed genus, or category. We demonstrate that even when trained on a single high-resolution mesh our method generates reasonable subdivisions for novel shapes.
  arXiv  Detail & Related papers  (2020-05-04T20:03:21Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.