# Gauge Equivariant Mesh CNNs: Anisotropic convolutions on geometric graphs

arXiv: https://arxiv.org/abs/2003.05425 · HTML: https://mlsanity.com/paper/2003.05425/

## Key facts

- Cited by 92 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Mar 2020.

## Built on by (most cited first, 30 of 92)

- [Neural Sheaf Diffusion: A Topological Perspective on Heterophily and Oversmoothing in GNNs](https://mlsanity.com/paper/2202.04579/) (2022)
- [Spherical Message Passing for 3D Graph Networks](https://mlsanity.com/paper/2102.05013/) (2021)
- [GTA: A Geometry-Aware Attention Mechanism for Multi-View Transformers](https://mlsanity.com/paper/2310.10375/) (2023)
- [Geometric Deep Learning and Equivariant Neural Networks](https://mlsanity.com/paper/2105.13926/) (2021)
- [When Physics Meets Machine Learning: A Survey of Physics-Informed Machine Learning](https://mlsanity.com/paper/2203.16797/) (2022)
- [Geometric Algebra Transformer](https://mlsanity.com/paper/2305.18415/) (2023)
- [Fast, Expressive SE(n) Equivariant Networks through Weight-Sharing in Position-Orientation Space](https://mlsanity.com/paper/2310.02970/) (2023)
- [The Surprising Effectiveness of Equivariant Models in Domains with Latent Symmetry](https://mlsanity.com/paper/2211.09231/) (2022)
- [Position: Categorical Deep Learning is an Algebraic Theory of All Architectures](https://mlsanity.com/paper/2402.15332/) (2024)
- [Mesh Neural Networks for SE(3)-Equivariant Hemodynamics Estimation on the Artery Wall](https://mlsanity.com/paper/2212.05023/) (2022)
- [SE(3)-Equivariant Attention Networks for Shape Reconstruction in Function Space](https://mlsanity.com/paper/2204.02394/) (2022)
- [Clifford-Steerable Convolutional Neural Networks](https://mlsanity.com/paper/2402.14730/) (2024)
- [Roto-translated Local Coordinate Frames For Interacting Dynamical Systems](https://mlsanity.com/paper/2110.14961/) (2021)
- [Steerable Partial Differential Operators for Equivariant Neural Networks](https://mlsanity.com/paper/2106.10163/) (2021)
- [Equi-Tuning: Group Equivariant Fine-Tuning of Pretrained Models](https://mlsanity.com/paper/2210.06475/) (2022)
- [Tangent Bundle Convolutional Learning: from Manifolds to Cellular Sheaves and Back](https://mlsanity.com/paper/2303.11323/) (2023)
- [Machine Learning for Partial Differential Equations](https://mlsanity.com/paper/2303.17078/) (2023)
- [Scaling Spherical CNNs](https://mlsanity.com/paper/2306.05420/) (2023)
- [Unified Fourier-based Kernel and Nonlinearity Design for Equivariant Networks on Homogeneous Spaces](https://mlsanity.com/paper/2206.08362/) (2022)
- [Vector-valued Gaussian Processes on Riemannian Manifolds via Gauge Independent Projected Kernels](https://mlsanity.com/paper/2110.14423/) (2021)
- [4D Panoptic Segmentation as Invariant and Equivariant Field Prediction](https://mlsanity.com/paper/2303.15651/) (2023)
- [SPANet: Generalized Permutationless Set Assignment for Particle Physics using Symmetry Preserving Attention](https://mlsanity.com/paper/2106.03898/) (2021)
- [Large-scale graph representation learning with very deep GNNs and self-supervision](https://mlsanity.com/paper/2107.09422/) (2021)
- [Equivariance Discovery by Learned Parameter-Sharing](https://mlsanity.com/paper/2204.03640/) (2022)
- [Learnable Polyphase Sampling for Shift Invariant and Equivariant Convolutional Networks](https://mlsanity.com/paper/2210.08001/) (2022)
- [Equivariant Networks for Pixelized Spheres](https://mlsanity.com/paper/2106.06662/) (2021)
- [Making Vision Transformers Truly Shift-Equivariant](https://mlsanity.com/paper/2305.16316/) (2023)
- [Truly Scale-Equivariant Deep Nets with Fourier Layers](https://mlsanity.com/paper/2311.02922/) (2023)
- [Field Convolutions for Surface CNNs](https://mlsanity.com/paper/2104.03916/) (2021)
- [Graph Neural Network with Local Frame for Molecular Potential Energy Surface](https://mlsanity.com/paper/2208.00716/) (2022)

## Abstract (arXiv)

A common approach to define convolutions on meshes is to interpret them as a graph and apply graph convolutional networks (GCNs). Such GCNs utilize isotropic kernels and are therefore insensitive to the relative orientation of vertices and thus to the geometry of the mesh as a whole. We propose Gauge Equivariant Mesh CNNs which generalize GCNs to apply anisotropic gauge equivariant kernels. Since the resulting features carry orientation information, we introduce a geometric message passing scheme defined by parallel transporting features over mesh edges. Our experiments validate the significantly improved expressivity of the proposed model over conventional GCNs and other methods.