# Foundations of Equivariant Deep Learning: Unifying Graph and Sheaf Neural Networks

arXiv: https://arxiv.org/abs/2607.03798 · HTML: https://mlsanity.com/paper/2607.03798/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 44 references that are in the map.
- First posted on arXiv 4 Jul 2026.

## Builds on (most cited first, 22 of 22)

- [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)
- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (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)
- [SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networks](https://mlsanity.com/paper/2006.10503/) (2020)
- [General E(2)-Equivariant Steerable CNNs](https://mlsanity.com/paper/1911.08251/) (2019)
- [Invariant and Equivariant Graph Networks](https://mlsanity.com/paper/1812.09902/) (2018)
- [Steerable CNNs](https://mlsanity.com/paper/1612.08498/) (2016)
- [Neural Sheaf Diffusion: A Topological Perspective on Heterophily and Oversmoothing in GNNs](https://mlsanity.com/paper/2202.04579/) (2022)
- [Simplicial Neural Networks](https://mlsanity.com/paper/2010.03633/) (2020)
- [Topological Deep Learning: Going Beyond Graph Data](https://mlsanity.com/paper/2206.00606/) (2022)
- [Sheaf Neural Networks](https://mlsanity.com/paper/2012.06333/) (2020)
- [Sheaf Neural Networks with Connection Laplacians](https://mlsanity.com/paper/2206.08702/) (2022)
- [E(n) Equivariant Topological Neural Networks](https://mlsanity.com/paper/2405.15429/) (2024)
- [Bundle Neural Networks for message diffusion on graphs](https://mlsanity.com/paper/2405.15540/) (2024)
- [Copresheaf Topological Neural Networks: A Generalized Deep Learning Framework](https://mlsanity.com/paper/2505.21251/) (2025)
- [Categorical Equivariant Deep Learning: Category-Equivariant Neural Networks and Universal Approximation Theorems](https://mlsanity.com/paper/2511.18417/) (2025)
- [Learning with Category-Equivariant Architectures for Human Activity Recognition](https://mlsanity.com/paper/2511.01139/) (2025)
- [Learning with Category-Equivariant Representations for Human Activity Recognition](https://mlsanity.com/paper/2511.00900/) (2025)

## Built on by (most cited first, 1 of 1)

- [Equivariant Sheaf Neural Networks: Learning Geometric Transport on Graphs](https://mlsanity.com/paper/2608.28853/) (2026)

## Abstract (arXiv)

Symmetry is everywhere in nature and society. Geometric deep learning builds architectures respecting group symmetries, whereas topological deep learning organizes computation through cells, incidence relations, and local-to-global structure. In this paper, we extend geometric deep learning beyond simple group actions and unify it with topological deep learning. Specifically, we develop order-equivariant neural networks (OENN), which generalize standard graph message passing and sheaf neural networks via the theory of equivariant vector bundles over face posets (or face categories). We (i) characterize all linear order-equivariant maps, (ii) build OENN layers, and (iii) prove universal approximation theorems (UATs) for continuous order-equivariant maps, which are new results even when restricted to sheaf neural networks. We illustrate the framework on graph and sheaf models. Our results can also be seen as extending the known UAT for graph neural networks to a more general setting that subsumes sheaf neural networks as well. In the appendix, we show that OENN can be connected, via the action groupoid Grothendieck construction, to CENN (category-equivariant neural network), which gives the categorical general form of equivariant neural networks, allowing us to leverage categorical symmetry in data and extending geometric deep learning from groups of symmetries to categories of transformations.