# What is an equivariant neural network?

arXiv: https://arxiv.org/abs/2205.07362 · HTML: https://mlsanity.com/paper/2205.07362/

## Key facts

- Cited by 21 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 25 references that are in the map.
- First posted on arXiv 15 May 2022.

## Builds on (most cited first, 11 of 11)

- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networks](https://mlsanity.com/paper/2006.10503/) (2020)
- [Making Convolutional Networks Shift-Invariant Again](https://mlsanity.com/paper/1904.11486/) (2019)
- [3D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Data](https://mlsanity.com/paper/1807.02547/) (2018)
- [On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups](https://mlsanity.com/paper/1802.03690/) (2018)
- [A Practical Method for Constructing Equivariant Multilayer Perceptrons for Arbitrary Matrix Groups](https://mlsanity.com/paper/2104.09459/) (2021)
- [Clebsch-Gordan Nets: a Fully Fourier Space Spherical Convolutional Neural Network](https://mlsanity.com/paper/1806.09231/) (2018)
- [On the Universality of Invariant Networks](https://mlsanity.com/paper/1901.09342/) (2019)
- [Deep neural network solution of the electronic Schr"odinger equation](https://mlsanity.com/paper/1909.08423/) (2019)
- [Lorentz Group Equivariant Neural Network for Particle Physics](https://mlsanity.com/paper/2006.04780/) (2020)

## Built on by (most cited first, 21 of 21)

- [An Efficient Sparse Kernel Generator for O(3)-Equivariant Deep Networks](https://mlsanity.com/paper/2501.13986/) (2025)
- [Brauer's Group Equivariant Neural Networks](https://mlsanity.com/paper/2212.08630/) (2022)
- [Connecting Permutation Equivariant Neural Networks and Partition Diagrams](https://mlsanity.com/paper/2212.08648/) (2022)
- [Any-dimensional equivariant neural networks](https://mlsanity.com/paper/2306.06327/) (2023)
- [A Comparison Between Invariant and Equivariant Classical and Quantum Graph Neural Networks](https://mlsanity.com/paper/2311.18672/) (2023)
- [How Jellyfish Characterise Alternating Group Equivariant Neural Networks](https://mlsanity.com/paper/2301.10152/) (2023)
- [Geometry of Linear Neural Networks: Equivariance and Invariance under Permutation Groups](https://mlsanity.com/paper/2309.13736/) (2023)
- [Graph Automorphism Group Equivariant Neural Networks](https://mlsanity.com/paper/2307.07810/) (2023)
- [Lorentz-Equivariant Quantum Graph Neural Network for High-Energy Physics](https://mlsanity.com/paper/2411.01641/) (2024)
- [Equivariant neural networks and piecewise linear representation theory](https://mlsanity.com/paper/2408.00949/) (2024)
- [Split personalities in Bayesian Neural Networks: the case for full marginalisation](https://mlsanity.com/paper/2205.11151/) (2022)
- [VC dimensions of group convolutional neural networks](https://mlsanity.com/paper/2212.09507/) (2022)
- [A large language model-type architecture for high-dimensional molecular potential energy surfaces](https://mlsanity.com/paper/2412.03831/) (2024)
- [REViT: Roto-reflection Equivariant Convolutional Vision Transformer](https://mlsanity.com/paper/2606.25318/) (2026)
- [Discrete neural nets and polymorphic learning](https://mlsanity.com/paper/2308.00677/) (2023)
- [Compact Matrix Quantum Group Equivariant Neural Networks](https://mlsanity.com/paper/2311.06358/) (2023)
- [CNN-based Compressor Mass Flow Estimator in Industrial Aircraft Vapor Cycle System](https://mlsanity.com/paper/2406.17788/) (2024)
- [Lie-Equivariant Quantum Graph Neural Networks](https://mlsanity.com/paper/2411.15315/) (2024)
- [Precoder Learning by Leveraging Unitary Equivariance Property](https://mlsanity.com/paper/2503.09398/) (2025)
- [Feature Shift Localization Network](https://mlsanity.com/paper/2506.09101/) (2025)
- [Multi-objective optimization and quantum hybridization of equivariant deep learning interatomic potentials](https://mlsanity.com/paper/2602.16908/) (2026)

## Abstract (arXiv)

We explain equivariant neural networks, a notion underlying breakthroughs in machine learning from deep convolutional neural networks for computer vision to AlphaFold 2 for protein structure prediction, without assuming knowledge of equivariance or neural networks. The basic mathematical ideas are simple but are often obscured by engineering complications that come with practical realizations. We extract and focus on the mathematical aspects, and limit ourselves to a cursory treatment of the engineering issues at the end.