# Clifford Group Equivariant Simplicial Message Passing Networks

arXiv: https://arxiv.org/abs/2402.10011 · HTML: https://mlsanity.com/paper/2402.10011/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- Builds on 37 of its 52 references that are in the map.
- First posted on arXiv 15 Feb 2024.

## Builds on (most cited first, 30 of 37)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [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)
- [Fast Graph Representation Learning with PyTorch Geometric](https://mlsanity.com/paper/1903.02428/) (2019)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [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)
- [Graph Neural Networks for Social Recommendation](https://mlsanity.com/paper/1902.07243/) (2019)
- [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)
- [Interaction Networks for Learning about Objects, Relations and Physics](https://mlsanity.com/paper/1612.00222/) (2016)
- [Neural Relational Inference for Interacting Systems](https://mlsanity.com/paper/1802.04687/) (2018)
- [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)
- [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)
- [Can Graph Neural Networks Count Substructures?](https://mlsanity.com/paper/2002.04025/) (2020)
- [Geometric and Physical Quantities Improve E(3) Equivariant Message Passing](https://mlsanity.com/paper/2110.02905/) (2021)
- [Weisfeiler and Lehman Go Topological: Message Passing Simplicial Networks](https://mlsanity.com/paper/2103.03212/) (2021)
- [TorchMD-NET: Equivariant Transformers for Neural Network based Molecular Potentials](https://mlsanity.com/paper/2202.02541/) (2022)
- [Predicting multicellular function through multi-layer tissue networks](https://mlsanity.com/paper/1707.04638/) (2017)
- [GroupNet: Multiscale Hypergraph Neural Networks for Trajectory Prediction with Relational Reasoning](https://mlsanity.com/paper/2204.08770/) (2022)
- [EqMotion: Equivariant Multi-agent Motion Prediction with Invariant Interaction Reasoning](https://mlsanity.com/paper/2303.10876/) (2023)
- [Clifford Neural Layers for PDE Modeling](https://mlsanity.com/paper/2209.04934/) (2022)
- [Social Ways: Learning Multi-Modal Distributions of Pedestrian Trajectories with GANs](https://mlsanity.com/paper/1904.09507/) (2019)
- [Equivariant Graph Mechanics Networks with Constraints](https://mlsanity.com/paper/2203.06442/) (2022)
- [Expressiveness and Approximation Properties of Graph Neural Networks](https://mlsanity.com/paper/2204.04661/) (2022)
- [Geometrically Equivariant Graph Neural Networks: A Survey](https://mlsanity.com/paper/2202.07230/) (2022)
- [Collaborative Motion Prediction via Neural Motion Message Passing](https://mlsanity.com/paper/2003.06594/) (2020)

## Built on by (most cited first, 22 of 22)

- [Does equivariance matter at scale?](https://mlsanity.com/paper/2410.23179/) (2024)
- [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)
- [E(n) Equivariant Topological Neural Networks](https://mlsanity.com/paper/2405.15429/) (2024)
- [Beyond Euclid: An Illustrated Guide to Modern Machine Learning with Geometric, Topological, and Algebraic Structures](https://mlsanity.com/paper/2407.09468/) (2024)
- [Probing Equivariance and Symmetry Breaking in Convolutional Networks](https://mlsanity.com/paper/2501.01999/) (2025)
- [Controlled Generation with Equivariant Variational Flow Matching](https://mlsanity.com/paper/2506.18340/) (2025)
- [HOG-Diff: Higher-Order Guided Diffusion for Graph Generation](https://mlsanity.com/paper/2502.04308/) (2025)
- [Multivector Neurons: Better and Faster O(n)-Equivariant Clifford Graph Neural Networks](https://mlsanity.com/paper/2406.04052/) (2024)
- [A Clifford Algebraic Approach to E(n)-Equivariant High-order Graph Neural Networks](https://mlsanity.com/paper/2410.04692/) (2024)
- [Equivariant Neural Networks for General Linear Symmetries on Lie Algebras](https://mlsanity.com/paper/2510.22984/) (2025)
- [ILPO-NET: Network for the invariant recognition of arbitrary volumetric patterns in 3D](https://mlsanity.com/paper/2403.19612/) (2024)
- [E(n) Equivariant Message Passing Cellular Networks](https://mlsanity.com/paper/2406.03145/) (2024)
- [Clifford Group Equivariant Diffusion Models for 3D Molecular Generation](https://mlsanity.com/paper/2504.15773/) (2025)
- [Resolving Oversmoothing with Opinion Dissensus](https://mlsanity.com/paper/2501.19089/) (2025)
- [AdS-GNN -- a Conformally Equivariant Graph Neural Network](https://mlsanity.com/paper/2505.12880/) (2025)
- [GLGENN: A Novel Parameter-Light Equivariant Neural Networks Architecture Based on Clifford Geometric Algebras](https://mlsanity.com/paper/2506.09625/) (2025)
- [Geometric Algebra Meets Cartesian Tensors: Higher-Order Equivariance for Interatomic Potentials](https://mlsanity.com/paper/2606.29584/) (2026)
- [ART: Adaptive Reasoning Trees for Explainable Claim Verification](https://mlsanity.com/paper/2601.05455/) (2026)
- [Weisfeiler Lehman Test on Combinatorial Complexes: Generalized Expressive Power of Topological Neural Networks](https://mlsanity.com/paper/2605.00725/) (2026)
- [Learning Discrete Riemannian Metrics for Physical Fields with Cochain-Frame Equivarianc](https://mlsanity.com/paper/2608.14556/) (2026)
- [Clifford Sheaf Neural Networks](https://mlsanity.com/paper/2610.01322/) (2026)

## Abstract (arXiv)

We introduce Clifford Group Equivariant Simplicial Message Passing Networks, a method for steerable E(n)-equivariant message passing on simplicial complexes. Our method integrates the expressivity of Clifford group-equivariant layers with simplicial message passing, which is topologically more intricate than regular graph message passing. Clifford algebras include higher-order objects such as bivectors and trivectors, which express geometric features (e.g., areas, volumes) derived from vectors. Using this knowledge, we represent simplex features through geometric products of their vertices. To achieve efficient simplicial message passing, we share the parameters of the message network across different dimensions. Additionally, we restrict the final message to an aggregation of the incoming messages from different dimensions, leading to what we term shared simplicial message passing. Experimental results show that our method is able to outperform both equivariant and simplicial graph neural networks on a variety of geometric tasks.