mlsanityData to 14 Oct 2026

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

arXiv 2301.05638 ↗First posted 13 Jan 2023hep-ph cs.LG physics.data-ancited by 22

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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.

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

31 of 58 references.

References · 31

  1. 2019PyTorch: An Imperative Style, High-Performance Deep Learning Librarycited by 20,305
  2. 2019AI Feynman: a Physics-Inspired Method for Symbolic Regressioncited by 548
  3. 2020Lagrangian Neural Networkscited by 398
  4. 2020Discovering Symbolic Models from Deep Learning with Inductive Biasescited by 344
  5. 2019Deep Learning for Symbolic Mathematicscited by 270
  6. 2022End-to-end symbolic regression with transformerscited by 203
  7. 2020Lorentz Group Equivariant Neural Network for Particle Physicscited by 121
  8. 2018Discovering physical concepts with neural networkscited by 117
  9. 2020Equivariant flow-based sampling for lattice gauge theorycited by 103
  10. 2020AI Poincar'e: Machine Learning Conservation Laws from Trajectoriescited by 77
  11. 2022Rediscovering orbital mechanics with machine learningcited by 71
  12. 2022Deep Symbolic Regression for Recurrent Sequencescited by 54
19 more
  1. 2022The Lie Derivative for Measuring Learned Equivariancecited by 54
  2. 2022Rethinking Symbolic Regression Datasets and Benchmarks for Scientific Discoverycited by 46
  3. 2020Detecting Symmetries with Neural Networkscited by 34
  4. 2020Discovering Symmetry Invariants and Conserved Quantities by Interpreting Siamese Neural Networkscited by 33
  5. 2021Machine-learning hidden symmetriescited by 31
  6. 2022LieGG: Studying Learned Lie Group Generatorscited by 31
  7. 2022Symmetry Group Equivariant Architectures for Physicscited by 23
  8. 2020Permutationless Many-Jet Event Reconstruction with Symmetry Preserving Attention Networkscited by 22
  9. 2021SPANet: Generalized Permutationless Set Assignment for Particle Physics using Symmetry Preserving Attentioncited by 18
  10. 2021Analytical Modelling of Exoplanet Transit Specroscopy with Dimensional Analysis and Symbolic Regressioncited by 18
  11. 2022Simplifying Polylogarithms with Machine Learningcited by 15
  12. 2022Lorentz group equivariant autoencoderscited by 15
  13. 2020Machine Learning Lie Structures & Applications to Physicscited by 13
  14. 2021Symmetry meets AIcited by 13
  15. 2022SYMBA: Symbolic Computation of Squared Amplitudes in High Energy Physics with Machine Learningcited by 13
  16. 2021Machine learning a manifoldcited by 9
  17. 2021Accelerating Understanding of Scientific Experiments with End to End Symbolic Regressioncited by 8
  18. 2022Is the Machine Smarter than the Theorist: Deriving Formulas for Particle Kinematics with Symbolic Regressioncited by 8
  19. 2022Symbolic Expression Transformer: A Computer Vision Approach for Symbolic Regressioncited by 7

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2023: 7; 2024: 7; 2025: 6; 2026: 2.

22 papers.

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  1. 2024LieRE: Lie Rotational Positional Encodingscited by 18
  2. 2024Learning Infinitesimal Generators of Continuous Symmetries from Datacited by 17
  3. 2025Interpretable Machine Learning in Physics: A Reviewcited by 14
  4. 2023Discovering Sparse Representations of Lie Groups with Machine Learningcited by 8
  5. 2023Accelerated Discovery of Machine-Learned Symmetries: Deriving the Exceptional Lie Groups G2, F4 and E6cited by 6
  6. 2023Identifying the Group-Theoretic Structure of Machine-Learned Symmetriescited by 6
  7. 2023A Comparison Between Invariant and Equivariant Classical and Quantum Graph Neural Networkscited by 6
  8. 2023Oracle-Preserving Latent Flowscited by 5
  9. 2025AtlasD: Automatic Local Symmetry Discoverycited by 5
  10. 2023The R-mAtrIx Netcited by 4
  11. 2024Lorentz-Equivariant Quantum Graph Neural Network for High-Energy Physicscited by 4
  12. 2025Learning finite symmetry groups of dynamical systems via equivariance detectioncited by 2
10 more
  1. 2024SymmetryLens: Unsupervised Symmetry Learning via Locality and Density Preservationcited by 1
  2. 2025RECON: Robust symmetry discovery via Explicit Canonical Orientation Normalizationcited by 1
  3. 2023A charge-preserving method for solving graph neural diffusion networkscited by 0
  4. 2024Optimal Potential Shaping on SE(3) via Neural ODEs on Lie Groupscited by 0
  5. 2024Normalized Space Alignment: A Versatile Metric for Representation Analysiscited by 0
  6. 2024Lie-Equivariant Quantum Graph Neural Networkscited by 0
  7. 2025Non-Gaussianities in Collider Metric Binningcited by 0
  8. 2025Deep Learning based discovery of Integrable Systemscited by 0
  9. 2026Blind Recovery of Latent Domains via Unsupervised Symmetry Discoverycited by 0
  10. 2026LieStoNet: Learning Lie Symmetries from Spatiotemporal Data for Stochastic Dynamical Systemscited by 0