# Covariance in Physics and Convolutional Neural Networks

arXiv: https://arxiv.org/abs/1906.02481 · HTML: https://mlsanity.com/paper/1906.02481/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Jun 2019.

## Built on by (most cited first, 12 of 12)

- [Geometric Deep Learning and Equivariant Neural Networks](https://mlsanity.com/paper/2105.13926/) (2021)
- [Lorentz-Equivariant Geometric Algebra Transformers for High-Energy Physics](https://mlsanity.com/paper/2405.14806/) (2024)
- [Homogeneous vector bundles and G-equivariant convolutional neural networks](https://mlsanity.com/paper/2105.05400/) (2021)
- [HEAL-SWIN: A Vision Transformer On The Sphere](https://mlsanity.com/paper/2307.07313/) (2023)
- [Towards fully covariant machine learning](https://mlsanity.com/paper/2301.13724/) (2023)
- [Geometrical aspects of lattice gauge equivariant convolutional neural networks](https://mlsanity.com/paper/2303.11448/) (2023)
- [Noncommutative Geometry of Computational Models and Uniformization for Framed Quiver Varieties](https://mlsanity.com/paper/2201.05900/) (2022)
- [Learning Chern Numbers of Topological Insulators with Gauge Equivariant Neural Networks](https://mlsanity.com/paper/2502.15376/) (2025)
- [Computing equivariant matrices on homogeneous spaces for Geometric Deep Learning and Automorphic Lie Algebras](https://mlsanity.com/paper/2303.07157/) (2023)
- [Quantum Finite Automata and Quiver Algebras](https://mlsanity.com/paper/2203.07597/) (2022)
- [Steerable Neural ODEs on Homogeneous Spaces](https://mlsanity.com/paper/2605.11133/) (2026)
- [Deep neural networks as lattice gauge theories](https://mlsanity.com/paper/2608.19331/) (2026)

## Abstract (arXiv)

In this proceeding we give an overview of the idea of covariance (or equivariance) featured in the recent development of convolutional neural networks (CNNs). We study the similarities and differences between the use of covariance in theoretical physics and in the CNN context. Additionally, we demonstrate that the simple assumption of covariance, together with the required properties of locality, linearity and weight sharing, is sufficient to uniquely determine the form of the convolution.