mlsanityData to 7 Oct 2026

Automatic Symmetry Discovery with Lie Algebra Convolutional Network

arXiv 2109.07103 ↗First posted 15 Sep 2021cs.LG cs.AI math.GRcited by 82

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Abstract (arXiv)

Existing equivariant neural networks require prior knowledge of the symmetry group and discretization for continuous groups. We propose to work with Lie algebras (infinitesimal generators) instead of Lie groups. Our model, the Lie algebra convolutional network (L-conv) can automatically discover symmetries and does not require discretization of the group. We show that L-conv can serve as a building block to construct any group equivariant feedforward architecture. Both CNNs and Graph Convolutional Networks can be expressed as L-conv with appropriate groups. We discover direct connections between L-conv and physics: (1) group invariant loss generalizes field theory (2) Euler-Lagrange equation measures the robustness, and (3) equivariance leads to conservation laws and Noether current.These connections open up new avenues for designing more general equivariant networks and applying them to important problems in physical sciences

Key facts. Cited by 82 papers in the map (data through 7 Oct 2026). Builds on 28 of its 49 references that are in the map. First posted on arXiv 15 Sep 2021.

Builds on

28 of 49 references.

References · 28

  1. 2016Semi-Supervised Classification with Graph Convolutional Networkscited by 11,282
  2. 2017Neural Message Passing for Quantum Chemistrycited by 3,460
  3. 2017Deep Setscited by 1,394
  4. 2016Group Equivariant Convolutional Networkscited by 1,154
  5. 2021Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gaugescited by 1,131
  6. 2018Spherical CNNscited by 452
  7. 2019General E(2)-Equivariant Steerable CNNscited by 441
  8. 20183D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Datacited by 348
  9. 2018On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groupscited by 296
  10. 2016Harmonic Networks: Deep Translation and Rotation Equivariancecited by 293
  11. 2019Gauge Equivariant Convolutional Networks and the Icosahedral CNNcited by 262
  12. 2020Generalizing Convolutional Neural Networks for Equivariance to Lie Groups on Arbitrary Continuous Datacited by 248
16 more
  1. 2018A General Theory of Equivariant CNNs on Homogeneous Spacescited by 223
  2. 2017Learning Steerable Filters for Rotation Equivariant CNNscited by 193
  3. 2021A Practical Method for Constructing Equivariant Multilayer Perceptrons for Arbitrary Matrix Groupscited by 185
  4. 2018Clebsch-Gordan Nets: a Fully Fourier Space Spherical Convolutional Neural Networkcited by 168
  5. 2017Equivariance Through Parameter-Sharingcited by 139
  6. 2020Incorporating Symmetry into Deep Dynamics Models for Improved Generalizationcited by 123
  7. 2020Lorentz Group Equivariant Neural Network for Particle Physicscited by 120
  8. 2020On Learning Sets of Symmetric Elementscited by 97
  9. 2020Neural Mechanics: Symmetry and Broken Conservation Laws in Deep Learning Dynamicscited by 93
  10. 2018Random Walks on Simplicial Complexes and the normalized Hodge 1-Laplaciancited by 84
  11. 2020Meta-Learning Symmetries by Reparameterizationcited by 75
  12. 2018Deep Models of Interactions Across Setscited by 68
  13. 2020Learning Invariances in Neural Networkscited by 52
  14. 2020Universal Equivariant Multilayer Perceptronscited by 49
  15. 2014Learning the Irreducible Representations of Commutative Lie Groupscited by 41
  16. 2020Disentangling by Subspace Diffusioncited by 28

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 1; 2022: 17; 2023: 22; 2024: 13; 2025: 15; 2026: 14.

82 papers.

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  1. 2022On scientific understanding with artificial intelligencecited by 68
  2. 2022Learning Symmetric Embeddings for Equivariant World Modelscited by 54
  3. 2021Learning Partial Equivariances from Datacited by 44
  4. 2023Generative Adversarial Symmetry Discoverycited by 40
  5. 2022The Surprising Effectiveness of Equivariant Models in Domains with Latent Symmetrycited by 32
  6. 2023Latent Space Symmetry Discoverycited by 29
  7. 2022LieGG: Studying Learned Lie Group Generatorscited by 28
  8. 2022Equivariant Networks for Crystal Structurescited by 24
  9. 2023Learning Layer-wise Equivariances Automatically using Gradientscited by 23
  10. 2023Learning to Extrapolate: A Transductive Approachcited by 19
  11. 2023Discovering Symmetry Breaking in Physical Systems with Relaxed Group Convolutioncited by 19
  12. 2022Homomorphism Autoencoder -- Learning Group Structured Representations from Observed Transitionscited by 18
38 more
  1. 2023A General Theory of Correct, Incorrect, and Extrinsic Equivariancecited by 18
  2. 2022Structure-preserving GANscited by 17
  3. 2024Learning Infinitesimal Generators of Continuous Symmetries from Datacited by 17
  4. 2023Regularizing Towards Soft Equivariance Under Mixed Symmetriescited by 16
  5. 2022Unsupervised Learning of Equivariant Structure from Sequencescited by 15
  6. 2024Symmetry-Informed Governing Equation Discoverycited by 14
  7. 2024Symmetry Discovery Beyond Affine Transformationscited by 14
  8. 2023On genuine invariance learning without weight-tyingcited by 13
  9. 2022FINDE: Neural Differential Equations for Finding and Preserving Invariant Quantitiescited by 12
  10. 2023Almost Equivariance via Lie Algebra Convolutionscited by 10
  11. 2024A Generative Model of Symmetry Transformationscited by 10
  12. 2023Exploiting Code Symmetries for Learning Program Semanticscited by 9
  13. 2023Policy Gradient Methods in the Presence of Symmetries and State Abstractionscited by 8
  14. 2024Symmetry Discovery for Different Data Typescited by 7
  15. 2023Hamiltonian Neural Networks with Automatic Symmetry Detectioncited by 6
  16. 2023Learning Lie Group Symmetry Transformations with Neural Networkscited by 6
  17. 2023A Unified Framework for Discovering Discrete Symmetriescited by 6
  18. 2024Neural Isometries: Taming Transformations for Equivariant MLcited by 6
  19. 2025Diffusion Generative Modeling on Lie Group Representationscited by 6
  20. 2025Achieving Approximate Symmetry Is Exponentially Easier than Exact Symmetrycited by 6
  21. 2022Discrete Lagrangian Neural Networks with Automatic Symmetry Discoverycited by 5
  22. 2022Physics-guided Data Augmentation for Learning the Solution Operator of Linear Differential Equationscited by 5
  23. 2024Learning Symmetries via Weight-Sharing with Doubly Stochastic Tensorscited by 5
  24. 2025Explicit Discovery of Nonlinear Symmetries from Dynamic Datacited by 5
  25. 2022Learning Symmetric Rules with SATNetcited by 4
  26. 2023Using and Abusing Equivariancecited by 4
  27. 2025AtlasD: Automatic Local Symmetry Discoverycited by 4
  28. 2025Governing Equation Discovery from Data Based on Differential Invariantscited by 4
  29. 2022Homomorphic Self-Supervised Learningcited by 3
  30. 2023Non-parametric Hypothesis Tests for Distributional Group Symmetrycited by 3
  31. 2024ILPO-NET: Network for the invariant recognition of arbitrary volumetric patterns in 3Dcited by 3
  32. 2025Discovering Symmetry Groups with Flow Matchingcited by 3
  33. 2026Data Augmentation: A Fourier Analysis Perspectivecited by 3
  34. 2023Scale-Rotation-Equivariant Lie Group Convolution Neural Networks (Lie Group-CNNs)cited by 2
  35. 2023Neural Discovery of Permutation Subgroupscited by 2
  36. 2024CFASL: Composite Factor-Aligned Symmetry Learning for Disentanglement in Variational AutoEncodercited by 2
  37. 2025MatrixNet: Learning over symmetry groups using learned group representationscited by 2
  38. 2025Learning finite symmetry groups of dynamical systems via equivariance detectioncited by 2

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