# Deep Learning Symmetries and Their Lie Groups, Algebras, and Subalgebras from First Principles

arXiv: https://arxiv.org/abs/2301.05638 · HTML: https://mlsanity.com/paper/2301.05638/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- Builds on 31 of its 58 references that are in the map.
- First posted on arXiv 13 Jan 2023.

## Builds on (most cited first, 30 of 31)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [AI Feynman: a Physics-Inspired Method for Symbolic Regression](https://mlsanity.com/paper/1905.11481/) (2019)
- [Lagrangian Neural Networks](https://mlsanity.com/paper/2003.04630/) (2020)
- [Discovering Symbolic Models from Deep Learning with Inductive Biases](https://mlsanity.com/paper/2006.11287/) (2020)
- [Deep Learning for Symbolic Mathematics](https://mlsanity.com/paper/1912.01412/) (2019)
- [End-to-end symbolic regression with transformers](https://mlsanity.com/paper/2204.10532/) (2022)
- [Lorentz Group Equivariant Neural Network for Particle Physics](https://mlsanity.com/paper/2006.04780/) (2020)
- [Discovering physical concepts with neural networks](https://mlsanity.com/paper/1807.10300/) (2018)
- [Equivariant flow-based sampling for lattice gauge theory](https://mlsanity.com/paper/2003.06413/) (2020)
- [AI Poincar'e: Machine Learning Conservation Laws from Trajectories](https://mlsanity.com/paper/2011.04698/) (2020)
- [Rediscovering orbital mechanics with machine learning](https://mlsanity.com/paper/2202.02306/) (2022)
- [Deep Symbolic Regression for Recurrent Sequences](https://mlsanity.com/paper/2201.04600/) (2022)
- [The Lie Derivative for Measuring Learned Equivariance](https://mlsanity.com/paper/2210.02984/) (2022)
- [Rethinking Symbolic Regression Datasets and Benchmarks for Scientific Discovery](https://mlsanity.com/paper/2206.10540/) (2022)
- [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)
- [Symmetry Group Equivariant Architectures for Physics](https://mlsanity.com/paper/2203.06153/) (2022)
- [Permutationless Many-Jet Event Reconstruction with Symmetry Preserving Attention Networks](https://mlsanity.com/paper/2010.09206/) (2020)
- [SPANet: Generalized Permutationless Set Assignment for Particle Physics using Symmetry Preserving Attention](https://mlsanity.com/paper/2106.03898/) (2021)
- [Analytical Modelling of Exoplanet Transit Specroscopy with Dimensional Analysis and Symbolic Regression](https://mlsanity.com/paper/2112.11600/) (2021)
- [Simplifying Polylogarithms with Machine Learning](https://mlsanity.com/paper/2206.04115/) (2022)
- [Lorentz group equivariant autoencoders](https://mlsanity.com/paper/2212.07347/) (2022)
- [Machine Learning Lie Structures & Applications to Physics](https://mlsanity.com/paper/2011.00871/) (2020)
- [Symmetry meets AI](https://mlsanity.com/paper/2103.06115/) (2021)
- [SYMBA: Symbolic Computation of Squared Amplitudes in High Energy Physics with Machine Learning](https://mlsanity.com/paper/2206.08901/) (2022)
- [Machine learning a manifold](https://mlsanity.com/paper/2112.07673/) (2021)
- [Accelerating Understanding of Scientific Experiments with End to End Symbolic Regression](https://mlsanity.com/paper/2112.04023/) (2021)
- [Is the Machine Smarter than the Theorist: Deriving Formulas for Particle Kinematics with Symbolic Regression](https://mlsanity.com/paper/2211.08420/) (2022)

## Built on by (most cited first, 22 of 22)

- [LieRE: Lie Rotational Positional Encodings](https://mlsanity.com/paper/2406.10322/) (2024)
- [Learning Infinitesimal Generators of Continuous Symmetries from Data](https://mlsanity.com/paper/2410.21853/) (2024)
- [Interpretable Machine Learning in Physics: A Review](https://mlsanity.com/paper/2503.23616/) (2025)
- [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)
- [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)
- [Oracle-Preserving Latent Flows](https://mlsanity.com/paper/2302.00806/) (2023)
- [AtlasD: Automatic Local Symmetry Discovery](https://mlsanity.com/paper/2504.10777/) (2025)
- [The R-mAtrIx Net](https://mlsanity.com/paper/2304.07247/) (2023)
- [Lorentz-Equivariant Quantum Graph Neural Network for High-Energy Physics](https://mlsanity.com/paper/2411.01641/) (2024)
- [Learning finite symmetry groups of dynamical systems via equivariance detection](https://mlsanity.com/paper/2503.03014/) (2025)
- [SymmetryLens: Unsupervised Symmetry Learning via Locality and Density Preservation](https://mlsanity.com/paper/2410.05232/) (2024)
- [RECON: Robust symmetry discovery via Explicit Canonical Orientation Normalization](https://mlsanity.com/paper/2505.13289/) (2025)
- [A charge-preserving method for solving graph neural diffusion networks](https://mlsanity.com/paper/2312.10279/) (2023)
- [Optimal Potential Shaping on SE(3) via Neural ODEs on Lie Groups](https://mlsanity.com/paper/2401.15107/) (2024)
- [Normalized Space Alignment: A Versatile Metric for Representation Analysis](https://mlsanity.com/paper/2411.04512/) (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)
- [Deep Learning based discovery of Integrable Systems](https://mlsanity.com/paper/2503.10469/) (2025)
- [Blind Recovery of Latent Domains via Unsupervised Symmetry Discovery](https://mlsanity.com/paper/2606.17782/) (2026)
- [LieStoNet: Learning Lie Symmetries from Spatiotemporal Data for Stochastic Dynamical Systems](https://mlsanity.com/paper/2608.01582/) (2026)

## Abstract (arXiv)

We design a deep-learning algorithm for the discovery and identification of the continuous group of symmetries present in a labeled dataset. We use fully connected neural networks to model the symmetry transformations and the corresponding generators. We construct loss functions that ensure that the applied transformations are symmetries and that the corresponding set of generators forms a closed (sub)algebra. Our procedure is validated with several examples illustrating different types of conserved quantities preserved by symmetry. In the process of deriving the full set of symmetries, we analyze the complete subgroup structure of the rotation groups SO(2), SO(3), and SO(4), and of the Lorentz group SO(1,3). Other examples include squeeze mapping, piecewise discontinuous labels, and SO(10), demonstrating that our method is completely general, with many possible applications in physics and data science. Our study also opens the door for using a machine learning approach in the mathematical study of Lie groups and their properties.