# A Clifford Algebraic Approach to E(n)-Equivariant High-order Graph Neural Networks

arXiv: https://arxiv.org/abs/2410.04692 · HTML: https://mlsanity.com/paper/2410.04692/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 45 of its 65 references that are in the map.
- First posted on arXiv 7 Oct 2024.

## Builds on (most cited first, 30 of 45)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [SGDR: Stochastic Gradient Descent with Warm Restarts](https://mlsanity.com/paper/1608.03983/) (2016)
- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering](https://mlsanity.com/paper/1606.09375/) (2016)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Spectral Networks and Locally Connected Networks on Graphs](https://mlsanity.com/paper/1312.6203/) (2013)
- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [Weisfeiler and Leman Go Neural: Higher-order Graph Neural Networks](https://mlsanity.com/paper/1810.02244/) (2018)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentials](https://mlsanity.com/paper/2101.03164/) (2021)
- [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)
- [Directional Message Passing for Molecular Graphs](https://mlsanity.com/paper/2003.03123/) (2020)
- [MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields](https://mlsanity.com/paper/2206.07697/) (2022)
- [Equivariant message passing for the prediction of tensorial properties and molecular spectra](https://mlsanity.com/paper/2102.03150/) (2021)
- [Provably Powerful Graph Networks](https://mlsanity.com/paper/1905.11136/) (2019)
- [Invariant and Equivariant Graph Networks](https://mlsanity.com/paper/1812.09902/) (2018)
- [GemNet: Universal Directional Graph Neural Networks for Molecules](https://mlsanity.com/paper/2106.08903/) (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)
- [On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups](https://mlsanity.com/paper/1802.03690/) (2018)
- [Cormorant: Covariant Molecular Neural Networks](https://mlsanity.com/paper/1906.04015/) (2019)
- [Learning Local Equivariant Representations for Large-Scale Atomistic Dynamics](https://mlsanity.com/paper/2204.05249/) (2022)
- [Equivariant Flows: Exact Likelihood Generative Learning for Symmetric Densities](https://mlsanity.com/paper/2006.02425/) (2020)
- [Generalizing Convolutional Neural Networks for Equivariance to Lie Groups on Arbitrary Continuous Data](https://mlsanity.com/paper/2002.12880/) (2020)
- [Geometric and Physical Quantities Improve E(3) Equivariant Message Passing](https://mlsanity.com/paper/2110.02905/) (2021)
- [TorchMD-NET: Equivariant Transformers for Neural Network based Molecular Potentials](https://mlsanity.com/paper/2202.02541/) (2022)
- [On the Universality of Invariant Networks](https://mlsanity.com/paper/1901.09342/) (2019)
- [The Design Space of E(3)-Equivariant Atom-Centered Interatomic Potentials](https://mlsanity.com/paper/2205.06643/) (2022)

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

- [On Linear Mode Connectivity of Mixture-of-Experts Architectures](https://mlsanity.com/paper/2509.11348/) (2025)
- [Distance-Based Tree-Sliced Wasserstein Distance](https://mlsanity.com/paper/2503.11050/) (2025)
- [Tree-Sliced Wasserstein Distance with Nonlinear Projection](https://mlsanity.com/paper/2505.00968/) (2025)
- [Spherical Tree-Sliced Wasserstein Distance](https://mlsanity.com/paper/2503.11249/) (2025)
- [Quasi-Equivariant Metanetworks](https://mlsanity.com/paper/2604.23720/) (2026)

## Abstract (arXiv)

Designing neural network architectures that can handle data symmetry is crucial. This is especially important for geometric graphs whose properties are equivariance under Euclidean transformations. Current equivariant graph neural networks (EGNNs), particularly those using message passing, have a limitation in expressive power. Recent high-order graph neural networks can overcome this limitation, yet they lack equivariance properties, representing a notable drawback in certain applications in chemistry and physical sciences. In this paper, we introduce the Clifford Group Equivariant Graph Neural Networks (CG-EGNNs), a novel EGNN that enhances high-order message passing by integrating high-order local structures in the context of Clifford algebras. As a key benefit of using Clifford algebras, CG-EGNN can learn functions that capture equivariance from positional features. By adopting the high-order message passing mechanism, CG-EGNN gains richer information from neighbors, thus improving model performance. Furthermore, we establish the universality property of the k-hop message passing framework, showcasing greater expressive power of CG-EGNNs with additional k-hop message passing mechanism. We empirically validate that CG-EGNNs outperform previous methods on various benchmarks including n-body, CMU motion capture, and MD17, highlighting their effectiveness in geometric deep learning.