mlsanityData to 14 Oct 2026

Explicit Discovery of Nonlinear Symmetries from Dynamic Data

arXiv 2510.01855 ↗First posted 2 Oct 2025cs.LGcited by 6

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Abstract (arXiv)

Symmetry is widely applied in problems such as the design of equivariant networks and the discovery of governing equations, but in complex scenarios, it is not known in advance. Most previous symmetry discovery methods are limited to linear symmetries, and recent attempts to discover nonlinear symmetries fail to explicitly get the Lie algebra subspace. In this paper, we propose LieNLSD, which is, to our knowledge, the first method capable of determining the number of infinitesimal generators with nonlinear terms and their explicit expressions. We specify a function library for the infinitesimal group action and aim to solve for its coefficient matrix, proving that its prolongation formula for differential equations, which governs dynamic data, is also linear with respect to the coefficient matrix. By substituting the central differences of the data and the Jacobian matrix of the trained neural network into the infinitesimal criterion, we get a system of linear equations for the coefficient matrix, which can then be solved using SVD. On top quark tagging and a series of dynamic systems, LieNLSD shows qualitative advantages over existing methods and improves the long rollout accuracy of neural PDE solvers by over 20% while applying to guide data augmentation. Code and data are available at https://github.com/hulx2002/LieNLSD.

Key facts. Cited by 6 papers in the map (data through 14 Oct 2026). Builds on 46 of its 70 references that are in the map. First posted on arXiv 2 Oct 2025.

Builds on

46 of 70 references.

References · 46

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34 more
  1. 2022Adan: Adaptive Nesterov Momentum Algorithm for Faster Optimizing Deep Modelscited by 174
  2. 2020Incorporating Symmetry into Deep Dynamics Models for Improved Generalizationcited by 126
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  4. 2022Approximately Equivariant Networks for Imperfectly Symmetric Dynamicscited by 97
  5. 2018Roto-Translation Covariant Convolutional Networks for Medical Image Analysiscited by 92
  6. 2021Automatic Symmetry Discovery with Lie Algebra Convolutional Networkcited by 83
  7. 2022Lie Point Symmetry Data Augmentation for Neural PDE Solverscited by 77
  8. 2020Meta-Learning Symmetries by Reparameterizationcited by 76
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  15. 2023Generative Adversarial Symmetry Discoverycited by 42
  16. 2020Dense Steerable Filter CNNs for Exploiting Rotational Symmetry in Histology Imagescited by 33
  17. 2023Latent Space Symmetry Discoverycited by 29
  18. 2023Lie Point Symmetry and Physics Informed Networkscited by 24
  19. 2024Lie Algebra Canonicalization: Equivariant Neural Operators under arbitrary Lie Groupscited by 24
  20. 2023Learning Layer-wise Equivariances Automatically using Gradientscited by 23
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  22. 2024Trivialized Momentum Facilitates Diffusion Generative Modeling on Lie Groupscited by 19
  23. 2019Scaling-Translation-Equivariant Networks with Decomposed Convolutional Filterscited by 18
  24. 2024Learning Infinitesimal Generators of Continuous Symmetries from Datacited by 17
  25. 2017Deep Rotation Equivariant Networkcited by 16
  26. 2024Symmetry-Informed Governing Equation Discoverycited by 15
  27. 2024Symmetry Discovery Beyond Affine Transformationscited by 15
  28. 2018Cross-Domain 3D Equivariant Image Embeddingscited by 12
  29. 2024Symmetry Discovery for Different Data Typescited by 8
  30. 2025Diffusion Generative Modeling on Lie Group Representationscited by 8
  31. 2023Color Equivariant Convolutional Networkscited by 7
  32. 2022Physics-guided Data Augmentation for Learning the Solution Operator of Linear Differential Equationscited by 5
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  34. 2025Generalized Lie Symmetries in Physics-Informed Neural Operatorscited by 2

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2026: 6.

6 papers.

  1. 2026Adaptive Symmetry Discovery for Dynamical System Identificationcited by 3
  2. 2026Efficient Learning and Symmetry Discovery under Exact Invariancescited by 3
  3. 2026LieStoNet: Learning Lie Symmetries from Spatiotemporal Data for Stochastic Dynamical Systemscited by 0
  4. 2026Discovering Symmetries in Neural Network Parameter Spacescited by 0
  5. 2026LieDiscover: Adaptive Symbolic Library Construction for Explicit Open-form Symmetry Discoverycited by 0
  6. 2026Learning Operators for Symmetry Discovery: From Lie Groups to Local Symmetry Geometrycited by 0