# Accelerated Discovery of Machine-Learned Symmetries: Deriving the Exceptional Lie Groups G2, F4 and E6

arXiv: https://arxiv.org/abs/2307.04891 · HTML: https://mlsanity.com/paper/2307.04891/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 46 references that are in the map.
- First posted on arXiv 10 Jul 2023.

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

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [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)
- [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)
- [Machine Learning Lie Structures & Applications to Physics](https://mlsanity.com/paper/2011.00871/) (2020)
- [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)
- [Oracle-Preserving Latent Flows](https://mlsanity.com/paper/2302.00806/) (2023)

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

- [Identifying the Group-Theoretic Structure of Machine-Learned Symmetries](https://mlsanity.com/paper/2309.07860/) (2023)
- [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)
- [SymmetryLens: Unsupervised Symmetry Learning via Locality and Density Preservation](https://mlsanity.com/paper/2410.05232/) (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)

## Abstract (arXiv)

Recent work has applied supervised deep learning to derive continuous symmetry transformations that preserve the data labels and to obtain the corresponding algebras of symmetry generators. This letter introduces two improved algorithms that significantly speed up the discovery of these symmetry transformations. The new methods are demonstrated by deriving the complete set of generators for the unitary groups U(n) and the exceptional Lie groups G2, F4, and E6. A third post-processing algorithm renders the found generators in sparse form. We benchmark the performance improvement of the new algorithms relative to the standard approach. Given the significant complexity of the exceptional Lie groups, our results demonstrate that this machine-learning method for discovering symmetries is completely general and can be applied to a wide variety of labeled datasets.