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E(3)-Equivariant Mesh Neural Networks

Thuan Trang, Nhat Khang Ngo, Daniel Levy, Thieu N. Vo, Siamak Ravanbakhsh, Truong Son Hy

TL;DR

This work introduces EMNN, a simple $E(3)$-equivariant mesh neural network that minimally extends the EGNN framework to incorporate triangle-face information and enable hierarchical, multi-scale interactions. By integrating face geometry through invariant and equivariant messages and leveraging cross-product based normals, EMNN achieves state-of-the-art or competitive accuracy on mesh tasks while maintaining fast runtimes and modest memory footprints. The combination of multi-channel vector features and a hierarchical pooling/unpooling strategy enables effective long-range communication on long-diameter mesh graphs. Empirically, EMNN outperforms more complex mesh-specific equivariant architectures (e.g., GEM-CNN, EMAN) on equivariant benchmarks and remains strong on non-equivariant baselines, demonstrating practical impact for geometry-aware learning on meshes.

Abstract

Triangular meshes are widely used to represent three-dimensional objects. As a result, many recent works have address the need for geometric deep learning on 3D mesh. However, we observe that the complexities in many of these architectures does not translate to practical performance, and simple deep models for geometric graphs are competitive in practice. Motivated by this observation, we minimally extend the update equations of E(n)-Equivariant Graph Neural Networks (EGNNs) (Satorras et al., 2021) to incorporate mesh face information, and further improve it to account for long-range interactions through hierarchy. The resulting architecture, Equivariant Mesh Neural Network (EMNN), outperforms other, more complicated equivariant methods on mesh tasks, with a fast run-time and no expensive pre-processing. Our implementation is available at https://github.com/HySonLab/EquiMesh

E(3)-Equivariant Mesh Neural Networks

TL;DR

This work introduces EMNN, a simple -equivariant mesh neural network that minimally extends the EGNN framework to incorporate triangle-face information and enable hierarchical, multi-scale interactions. By integrating face geometry through invariant and equivariant messages and leveraging cross-product based normals, EMNN achieves state-of-the-art or competitive accuracy on mesh tasks while maintaining fast runtimes and modest memory footprints. The combination of multi-channel vector features and a hierarchical pooling/unpooling strategy enables effective long-range communication on long-diameter mesh graphs. Empirically, EMNN outperforms more complex mesh-specific equivariant architectures (e.g., GEM-CNN, EMAN) on equivariant benchmarks and remains strong on non-equivariant baselines, demonstrating practical impact for geometry-aware learning on meshes.

Abstract

Triangular meshes are widely used to represent three-dimensional objects. As a result, many recent works have address the need for geometric deep learning on 3D mesh. However, we observe that the complexities in many of these architectures does not translate to practical performance, and simple deep models for geometric graphs are competitive in practice. Motivated by this observation, we minimally extend the update equations of E(n)-Equivariant Graph Neural Networks (EGNNs) (Satorras et al., 2021) to incorporate mesh face information, and further improve it to account for long-range interactions through hierarchy. The resulting architecture, Equivariant Mesh Neural Network (EMNN), outperforms other, more complicated equivariant methods on mesh tasks, with a fast run-time and no expensive pre-processing. Our implementation is available at https://github.com/HySonLab/EquiMesh
Paper Structure (27 sections, 15 equations, 2 figures, 7 tables)

This paper contains 27 sections, 15 equations, 2 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Related Works
  3. Geometric Graph Learning
  4. Graph-based Deep Learning for Mesh
  5. Mesh as Manifold
  6. Background
  7. Mesh Representation
  8. Geometric Feature on Meshes
  9. Equivariant Graph Neural Networks
  10. Methodology
  11. Multiple Vector Channels
  12. Analysis of Equivariance
  13. Hierarchical Interactions
  14. Experiments
  15. Equivariant Benchmarks
  16. ...and 12 more sections

Figures (2)

  • Figure 1: EMNN Layer - Multi-channels version: On the left-hand side, invariant quantities, such as node's features, distances between two coordinates, and triangle surface areas, are used to update edge and face messages, denoted as $m_{ij}$ and $m_{ijk}$, respectively. Subsequently, on the right-hand side, these messages are combined with equivariant quantities, such as node's coordinates, relative positions between two coordinates, and the triangle surface normal vector, to update the node coordinates.
  • Figure 2: EMNN Network architecture for classification and segmentation: for classification, features are input into the pooling layers to extract global features used for predicting the label of each sample. For segmentation, the global values are fed to unpooling layers to produce node features.