# Using Multiple Vector Channels Improves E(n)-Equivariant Graph Neural Networks

arXiv: https://arxiv.org/abs/2309.03139 · HTML: https://mlsanity.com/paper/2309.03139/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 29 references that are in the map.
- First posted on arXiv 6 Sep 2023.

## Builds on (most cited first, 25 of 25)

- [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)
- [SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networks](https://mlsanity.com/paper/2006.10503/) (2020)
- [SchNet: A continuous-filter convolutional neural network for modeling quantum interactions](https://mlsanity.com/paper/1706.08566/) (2017)
- [Equivariant message passing for the prediction of tensorial properties and molecular spectra](https://mlsanity.com/paper/2102.03150/) (2021)
- [Learning from Protein Structure with Geometric Vector Perceptrons](https://mlsanity.com/paper/2009.01411/) (2020)
- [Vector Neurons: A General Framework for SO(3)-Equivariant Networks](https://mlsanity.com/paper/2104.12229/) (2021)
- [Cormorant: Covariant Molecular Neural Networks](https://mlsanity.com/paper/1906.04015/) (2019)
- [Fast and Uncertainty-Aware Directional Message Passing for Non-Equilibrium Molecules](https://mlsanity.com/paper/2011.14115/) (2020)
- [Generalizing Convolutional Neural Networks for Equivariance to Lie Groups on Arbitrary Continuous Data](https://mlsanity.com/paper/2002.12880/) (2020)
- [The Open Catalyst 2020 (OC20) Dataset and Community Challenges](https://mlsanity.com/paper/2010.09990/) (2020)
- [Geometric and Physical Quantities Improve E(3) Equivariant Message Passing](https://mlsanity.com/paper/2110.02905/) (2021)
- [A Practical Method for Constructing Equivariant Multilayer Perceptrons for Arbitrary Matrix Groups](https://mlsanity.com/paper/2104.09459/) (2021)
- [Spherical Message Passing for 3D Graph Networks](https://mlsanity.com/paper/2102.05013/) (2021)
- [Frame Averaging for Invariant and Equivariant Network Design](https://mlsanity.com/paper/2110.03336/) (2021)
- [Equivariance with Learned Canonicalization Functions](https://mlsanity.com/paper/2211.06489/) (2022)
- [Rediscovering orbital mechanics with machine learning](https://mlsanity.com/paper/2202.02306/) (2022)
- [Equivariant Graph Mechanics Networks with Constraints](https://mlsanity.com/paper/2203.06442/) (2022)
- [Equivariant Flows: sampling configurations for multi-body systems with symmetric energies](https://mlsanity.com/paper/1910.00753/) (2019)
- [Relevance of Rotationally Equivariant Convolutions for Predicting Molecular Properties](https://mlsanity.com/paper/2008.08461/) (2020)
- [Conditional Neural Relational Inference for Interacting Systems](https://mlsanity.com/paper/2106.11083/) (2021)
- [The Open MatSci ML Toolkit: A Flexible Framework for Machine Learning in Materials Science](https://mlsanity.com/paper/2210.17484/) (2022)
- [A Simple and Universal Rotation Equivariant Point-cloud Network](https://mlsanity.com/paper/2203.01216/) (2022)
- [Geometric Algebra Attention Networks for Small Point Clouds](https://mlsanity.com/paper/2110.02393/) (2021)

## Built on by (most cited first, 10 of 10)

- [Weisfeiler Leman for Euclidean Equivariant Machine Learning](https://mlsanity.com/paper/2402.02484/) (2024)
- [E(3)-Equivariant Mesh Neural Networks](https://mlsanity.com/paper/2402.04821/) (2024)
- [On the Expressive Power of Sparse Geometric MPNNs](https://mlsanity.com/paper/2407.02025/) (2024)
- [Symmetry-Aware Generative Modeling through Learned Canonicalization](https://mlsanity.com/paper/2501.07773/) (2025)
- [Scaling Laws and Symmetry, Evidence from Neural Force Fields](https://mlsanity.com/paper/2510.09768/) (2025)
- [Frame-based Equivariant Diffusion Models for 3D Molecular Generation](https://mlsanity.com/paper/2509.19506/) (2025)
- [AMShortcut: An Inference- and Training-Efficient Inverse Design Model for Amorphous Materials](https://mlsanity.com/paper/2603.29812/) (2026)
- [AMGenC: Generating Charge Balanced Amorphous Materials](https://mlsanity.com/paper/2604.27613/) (2026)
- [Augmented Equivariant Mesh Networks for Anatomical Segmentation](https://mlsanity.com/paper/2605.08172/) (2026)
- [Equivariant Sheaf Neural Networks: Learning Geometric Transport on Graphs](https://mlsanity.com/paper/2608.28853/) (2026)

## Abstract (arXiv)

We present a natural extension to E(n)-equivariant graph neural networks that uses multiple equivariant vectors per node. We formulate the extension and show that it improves performance across different physical systems benchmark tasks, with minimal differences in runtime or number of parameters. The proposed multichannel EGNN outperforms the standard singlechannel EGNN on N-body charged particle dynamics, molecular property predictions, and predicting the trajectories of solar system bodies. Given the additional benefits and minimal additional cost of multi-channel EGNN, we suggest that this extension may be of practical use to researchers working in machine learning for the physical sciences