# Identifying the Group-Theoretic Structure of Machine-Learned Symmetries

arXiv: https://arxiv.org/abs/2309.07860 · HTML: https://mlsanity.com/paper/2309.07860/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 38 references that are in the map.
- First posted on arXiv 14 Sep 2023.

## Builds on (most cited first, 12 of 12)

- [Discovering physical concepts with neural networks](https://mlsanity.com/paper/1807.10300/) (2018)
- [AI Poincar'e: Machine Learning Conservation Laws from Trajectories](https://mlsanity.com/paper/2011.04698/) (2020)
- [Detecting Symmetries with Neural Networks](https://mlsanity.com/paper/2003.13679/) (2020)
- [Discovering Symmetry Invariants and Conserved Quantities by Interpreting Siamese Neural Networks](https://mlsanity.com/paper/2003.04299/) (2020)
- [Machine-learning hidden symmetries](https://mlsanity.com/paper/2109.09721/) (2021)
- [LieGG: Studying Learned Lie Group Generators](https://mlsanity.com/paper/2210.04345/) (2022)
- [Deep Learning Symmetries and Their Lie Groups, Algebras, and Subalgebras from First Principles](https://mlsanity.com/paper/2301.05638/) (2023)
- [Symmetry meets AI](https://mlsanity.com/paper/2103.06115/) (2021)
- [Machine learning a manifold](https://mlsanity.com/paper/2112.07673/) (2021)
- [Discovering Sparse Representations of Lie Groups with Machine Learning](https://mlsanity.com/paper/2302.05383/) (2023)
- [Accelerated Discovery of Machine-Learned Symmetries: Deriving the Exceptional Lie Groups G2, F4 and E6](https://mlsanity.com/paper/2307.04891/) (2023)
- [Oracle-Preserving Latent Flows](https://mlsanity.com/paper/2302.00806/) (2023)

## Built on by (most cited first, 6 of 6)

- [Interpretable Machine Learning in Physics: A Review](https://mlsanity.com/paper/2503.23616/) (2025)
- [A Comparison Between Invariant and Equivariant Classical and Quantum Graph Neural Networks](https://mlsanity.com/paper/2311.18672/) (2023)
- [Lorentz-Equivariant Quantum Graph Neural Network for High-Energy Physics](https://mlsanity.com/paper/2411.01641/) (2024)
- [Quantum Rationale-Aware Graph Contrastive Learning for Jet Discrimination](https://mlsanity.com/paper/2411.01642/) (2024)
- [Lie-Equivariant Quantum Graph Neural Networks](https://mlsanity.com/paper/2411.15315/) (2024)
- [Non-Gaussianities in Collider Metric Binning](https://mlsanity.com/paper/2503.03809/) (2025)

## Abstract (arXiv)

Deep learning was recently successfully used in deriving symmetry transformations that preserve important physics quantities. Being completely agnostic, these techniques postpone the identification of the discovered symmetries to a later stage. In this letter we propose methods for examining and identifying the group-theoretic structure of such machine-learned symmetries. We design loss functions which probe the subalgebra structure either during the deep learning stage of symmetry discovery or in a subsequent post-processing stage. We illustrate the new methods with examples from the U(n) Lie group family, obtaining the respective subalgebra decompositions. As an application to particle physics, we demonstrate the identification of the residual symmetries after the spontaneous breaking of non-Abelian gauge symmetries like SU(3) and SU(5) which are commonly used in model building.