# Multivector Neurons: Better and Faster O(n)-Equivariant Clifford Graph Neural Networks

arXiv: https://arxiv.org/abs/2406.04052 · HTML: https://mlsanity.com/paper/2406.04052/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 18 references that are in the map.
- First posted on arXiv 6 Jun 2024.

## Builds on (most cited first, 17 of 17)

- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [Directional Message Passing for Molecular Graphs](https://mlsanity.com/paper/2003.03123/) (2020)
- [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)
- [Equiformer: Equivariant Graph Attention Transformer for 3D Atomistic Graphs](https://mlsanity.com/paper/2206.11990/) (2022)
- [Geometric and Physical Quantities Improve E(3) Equivariant Message Passing](https://mlsanity.com/paper/2110.02905/) (2021)
- [Clifford Neural Layers for PDE Modeling](https://mlsanity.com/paper/2209.04934/) (2022)
- [PiFold: Toward effective and efficient protein inverse folding](https://mlsanity.com/paper/2209.12643/) (2022)
- [Clifford Group Equivariant Neural Networks](https://mlsanity.com/paper/2305.11141/) (2023)
- [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)
- [Geometric Clifford Algebra Networks](https://mlsanity.com/paper/2302.06594/) (2023)
- [Lorentz-Equivariant Geometric Algebra Transformers for High-Energy Physics](https://mlsanity.com/paper/2405.14806/) (2024)
- [Clifford-Steerable Convolutional Neural Networks](https://mlsanity.com/paper/2402.14730/) (2024)
- [Clifford Group Equivariant Simplicial Message Passing Networks](https://mlsanity.com/paper/2402.10011/) (2024)
- [Weisfeiler Leman for Euclidean Equivariant Machine Learning](https://mlsanity.com/paper/2402.02484/) (2024)
- [SidechainNet: An All-Atom Protein Structure Dataset for Machine Learning](https://mlsanity.com/paper/2010.08162/) (2020)

## Built on by (most cited first, 5 of 5)

- [Does equivariance matter at scale?](https://mlsanity.com/paper/2410.23179/) (2024)
- [Breaking the Bottlenecks: Scalable Diffusion Models for 3D Molecular Generation](https://mlsanity.com/paper/2601.08963/) (2026)
- [Unifying Graph Neural Networks Through a Common Layer Equation](https://mlsanity.com/paper/2608.16097/) (2026)
- [Equivariant Sheaf Neural Networks: Learning Geometric Transport on Graphs](https://mlsanity.com/paper/2608.28853/) (2026)
- [Clifford Sheaf Neural Networks](https://mlsanity.com/paper/2610.01322/) (2026)

## Abstract (arXiv)

Most current deep learning models equivariant to O(n) or SO(n) either consider mostly scalar information such as distances and angles or have a very high computational complexity. In this work, we test a few novel message passing graph neural networks (GNNs) based on Clifford multivectors, structured similarly to other prevalent equivariant models in geometric deep learning. Our approach leverages efficient invariant scalar features while simultaneously performing expressive learning on multivector representations, particularly through the use of the equivariant geometric product operator. By integrating these elements, our methods outperform established efficient baseline models on an N-Body simulation task and protein denoising task while maintaining a high efficiency. In particular, we push the state-of-the-art error on the N-body dataset to 0.0035 (averaged over 3 runs); an 8% improvement over recent methods. Our implementation is available on Github.