# E(n) Equivariant Topological Neural Networks

arXiv: https://arxiv.org/abs/2405.15429 · HTML: https://mlsanity.com/paper/2405.15429/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- Builds on 76 of its 128 references that are in the map.
- First posted on arXiv 24 May 2024.

## Builds on (most cited first, 30 of 76)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (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)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [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)
- [Deeper Insights into Graph Convolutional Networks for Semi-Supervised Learning](https://mlsanity.com/paper/1801.07606/) (2018)
- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [Link Prediction Based on Graph Neural Networks](https://mlsanity.com/paper/1802.09691/) (2018)
- [Hierarchical Graph Representation Learning with Differentiable Pooling](https://mlsanity.com/paper/1806.08804/) (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)
- [Hypergraph Neural Networks](https://mlsanity.com/paper/1809.09401/) (2018)
- [MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields](https://mlsanity.com/paper/2206.07697/) (2022)
- [Self-Attention Graph Pooling](https://mlsanity.com/paper/1904.08082/) (2019)
- [Cormorant: Covariant Molecular Neural Networks](https://mlsanity.com/paper/1906.04015/) (2019)
- [Generalizing Convolutional Neural Networks for Equivariance to Lie Groups on Arbitrary Continuous Data](https://mlsanity.com/paper/2002.12880/) (2020)
- [A General Theory of Equivariant CNNs on Homogeneous Spaces](https://mlsanity.com/paper/1811.02017/) (2018)
- [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)
- [Weisfeiler and Lehman Go Cellular: CW Networks](https://mlsanity.com/paper/2106.12575/) (2021)
- [Neural Sheaf Diffusion: A Topological Perspective on Heterophily and Oversmoothing in GNNs](https://mlsanity.com/paper/2202.04579/) (2022)
- [On Over-Squashing in Message Passing Neural Networks: The Impact of Width, Depth, and Topology](https://mlsanity.com/paper/2302.02941/) (2023)
- [Protein Representation Learning by Geometric Structure Pretraining](https://mlsanity.com/paper/2203.06125/) (2022)
- [Simplicial Neural Networks](https://mlsanity.com/paper/2010.03633/) (2020)
- [LieTransformer: Equivariant self-attention for Lie Groups](https://mlsanity.com/paper/2012.10885/) (2020)

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

- [TopoBench: A Framework for Benchmarking Topological Deep Learning](https://mlsanity.com/paper/2406.06642/) (2024)
- [Beyond Euclid: An Illustrated Guide to Modern Machine Learning with Geometric, Topological, and Algebraic Structures](https://mlsanity.com/paper/2407.09468/) (2024)
- [TopoTune : A Framework for Generalized Combinatorial Complex Neural Networks](https://mlsanity.com/paper/2410.06530/) (2024)
- [Are High-Degree Representations Really Unnecessary in Equivariant Graph Neural Networks?](https://mlsanity.com/paper/2410.11443/) (2024)
- [MANTRA: The Manifold Triangulations Assemblage](https://mlsanity.com/paper/2410.02392/) (2024)
- [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)
- [HOPSE: Scalable Higher-Order Positional and Structural Encoder for Combinatorial Representations](https://mlsanity.com/paper/2505.15405/) (2025)
- [Equivariant Neural Networks for General Linear Symmetries on Lie Algebras](https://mlsanity.com/paper/2510.22984/) (2025)
- [Universally Invariant Learning in Equivariant GNNs](https://mlsanity.com/paper/2510.13169/) (2025)
- [On Universality of Deep Equivariant Networks](https://mlsanity.com/paper/2510.15814/) (2025)
- [FOLIAGE: Towards Physical Intelligence World Models Via Unbounded Surface Evolution](https://mlsanity.com/paper/2506.03173/) (2025)
- [Topotein: Topological Deep Learning for Protein Representation Learning](https://mlsanity.com/paper/2509.03885/) (2025)
- [Heat Kernel Goes Topological](https://mlsanity.com/paper/2507.12380/) (2025)
- [Foundations of Equivariant Deep Learning: Unifying Graph and Sheaf Neural Networks](https://mlsanity.com/paper/2607.03798/) (2026)
- [Directed Semi-Simplicial Learning with Applications to Brain Activity Decoding](https://mlsanity.com/paper/2505.17939/) (2025)
- [Weisfeiler Lehman Test on Combinatorial Complexes: Generalized Expressive Power of Topological Neural Networks](https://mlsanity.com/paper/2605.00725/) (2026)
- [Learning to Select Source Domains: Proxy-Rewarded Policy Optimization for Molecular OOD Generalization](https://mlsanity.com/paper/2605.13932/) (2026)
- [Adjusted Cup-Product Neural Layer](https://mlsanity.com/paper/2606.13568/) (2026)
- [Collapsed Effective Operators for Higher-order Structures](https://mlsanity.com/paper/2606.23517/) (2026)
- [Equivariant Sheaf Neural Networks: Learning Geometric Transport on Graphs](https://mlsanity.com/paper/2608.28853/) (2026)
- [EMERGE: Resolution-Agnostic Point Cloud Generation with Equivariant Graph-Based Diffusion](https://mlsanity.com/paper/2609.26039/) (2026)

## Abstract (arXiv)

Graph neural networks excel at modeling pairwise interactions, but they cannot flexibly accommodate higher-order interactions and features. Topological deep learning (TDL) has emerged recently as a promising tool for addressing this issue. TDL enables the principled modeling of arbitrary multi-way, hierarchical higher-order interactions by operating on combinatorial topological spaces, such as simplicial or cell complexes, instead of graphs. However, little is known about how to leverage geometric features such as positions and velocities for TDL. This paper introduces E(n)-Equivariant Topological Neural Networks (ETNNs), which are E(n)-equivariant message-passing networks operating on combinatorial complexes, formal objects unifying graphs, hypergraphs, simplicial, path, and cell complexes. ETNNs incorporate geometric node features while respecting rotation, reflection, and translation equivariance. Moreover, being TDL models, ETNNs are natively ready for settings with heterogeneous interactions. We provide a theoretical analysis to show the improved expressiveness of ETNNs over architectures for geometric graphs. We also show how E(n)-equivariant variants of TDL models can be directly derived from our framework. The broad applicability of ETNNs is demonstrated through two tasks of vastly different scales: i) molecular property prediction on the QM9 benchmark and ii) land-use regression for hyper-local estimation of air pollution with multi-resolution irregular geospatial data. The results indicate that ETNNs are an effective tool for learning from diverse types of richly structured data, as they match or surpass SotA equivariant TDL models with a significantly smaller computational burden, thus highlighting the benefits of a principled geometric inductive bias. Our implementation of ETNNs can be found at https://github.com/NSAPH-Projects/topological-equivariant-networks.