# E(3)-Equivariant Mesh Neural Networks

arXiv: https://arxiv.org/abs/2402.04821 · HTML: https://mlsanity.com/paper/2402.04821/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 34 of its 64 references that are in the map.
- First posted on arXiv 7 Feb 2024.

## Builds on (most cited first, 30 of 34)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [PointNet++: Deep Hierarchical Feature Learning on Point Sets in a Metric Space](https://mlsanity.com/paper/1706.02413/) (2017)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [Geometric deep learning on graphs and manifolds using mixture model CNNs](https://mlsanity.com/paper/1611.08402/) (2016)
- [Spherical CNNs](https://mlsanity.com/paper/1801.10130/) (2018)
- [Generating 3D faces using Convolutional Mesh Autoencoders](https://mlsanity.com/paper/1807.10267/) (2018)
- [Gauge Equivariant Convolutional Networks and the Icosahedral CNN](https://mlsanity.com/paper/1902.04615/) (2019)
- [Cormorant: Covariant Molecular Neural Networks](https://mlsanity.com/paper/1906.04015/) (2019)
- [Long Range Graph Benchmark](https://mlsanity.com/paper/2206.08164/) (2022)
- [Geometric and Physical Quantities Improve E(3) Equivariant Message Passing](https://mlsanity.com/paper/2110.02905/) (2021)
- [Tangent Convolutions for Dense Prediction in 3D](https://mlsanity.com/paper/1807.02443/) (2018)
- [DiffusionNet: Discretization Agnostic Learning on Surfaces](https://mlsanity.com/paper/2012.00888/) (2020)
- [Escape from Cells: Deep Kd-Networks for the Recognition of 3D Point Cloud Models](https://mlsanity.com/paper/1704.01222/) (2017)
- [SDM-NET: Deep Generative Network for Structured Deformable Mesh](https://mlsanity.com/paper/1908.04520/) (2019)
- [Learning shape correspondence with anisotropic convolutional neural networks](https://mlsanity.com/paper/1605.06437/) (2016)
- [MeshNet: Mesh Neural Network for 3D Shape Representation](https://mlsanity.com/paper/1811.11424/) (2018)
- [Gauge Equivariant Mesh CNNs: Anisotropic convolutions on geometric graphs](https://mlsanity.com/paper/2003.05425/) (2020)
- [SpiralNet++: A Fast and Highly Efficient Mesh Convolution Operator](https://mlsanity.com/paper/1911.05856/) (2019)
- [Coordinate Independent Convolutional Networks -- Isometry and Gauge Equivariant Convolutions on Riemannian Manifolds](https://mlsanity.com/paper/2106.06020/) (2021)
- [On the Connection Between MPNN and Graph Transformer](https://mlsanity.com/paper/2301.11956/) (2023)
- [Subdivision-Based Mesh Convolution Networks](https://mlsanity.com/paper/2106.02285/) (2021)
- [Covariant Compositional Networks For Learning Graphs](https://mlsanity.com/paper/1801.02144/) (2018)
- [TextureNet: Consistent Local Parametrizations for Learning from High-Resolution Signals on Meshes](https://mlsanity.com/paper/1812.00020/) (2018)
- [Primal-Dual Mesh Convolutional Neural Networks](https://mlsanity.com/paper/2010.12455/) (2020)
- [DualConvMesh-Net: Joint Geodesic and Euclidean Convolutions on 3D Meshes](https://mlsanity.com/paper/2004.01002/) (2020)
- [A Simple Approach to Intrinsic Correspondence Learning on Unstructured 3D Meshes](https://mlsanity.com/paper/1809.06664/) (2018)
- [Directional Message Passing on Molecular Graphs via Synthetic Coordinates](https://mlsanity.com/paper/2111.04718/) (2021)

## Built on by (most cited first, 9 of 9)

- [A Clifford Algebraic Approach to E(n)-Equivariant High-order Graph Neural Networks](https://mlsanity.com/paper/2410.04692/) (2024)
- [E(n) Equivariant Message Passing Cellular Networks](https://mlsanity.com/paper/2406.03145/) (2024)
- [TESGNN: Temporal Equivariant Scene Graph Neural Networks for Efficient and Robust Multi-View 3D Scene Understanding](https://mlsanity.com/paper/2411.10509/) (2024)
- [HodgeFormer: Transformers for Learnable Operators on Triangular Meshes through Data-Driven Hodge Matrices](https://mlsanity.com/paper/2509.01839/) (2025)
- [Thickness-aware E(3)-Equivariant 3D Mesh Neural Networks](https://mlsanity.com/paper/2505.21572/) (2025)
- [Drawback of Enforcing Equivariance and its Compensation via the Lens of Expressive Power](https://mlsanity.com/paper/2512.09673/) (2025)
- [PRIME: Protein Representation via Physics-Informed Multiscale Equivariant Hierarchies](https://mlsanity.com/paper/2605.01625/) (2026)
- [Augmented Equivariant Mesh Networks for Anatomical Segmentation](https://mlsanity.com/paper/2605.08172/) (2026)
- [EqGINO: Equivariant Geometry-Informed Fourier Neural Operators for 3D PDEs](https://mlsanity.com/paper/2606.03260/) (2026)

## Abstract (arXiv)

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