# Learning Infinitesimal Generators of Continuous Symmetries from Data arXiv: https://arxiv.org/abs/2410.21853 ยท HTML: https://mlsanity.com/paper/2410.21853/ ## Key facts - Cited by 17 papers in the map (data through 14 Oct 2026). - Builds on 17 of its 39 references that are in the map. - First posted on arXiv 29 Oct 2024. ## Builds on (most cited first, 17 of 17) - [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018) - [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020) - [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016) - [Equivariant Diffusion for Molecule Generation in 3D](https://mlsanity.com/paper/2203.17003/) (2022) - [Sensitivity and Generalization in Neural Networks: an Empirical Study](https://mlsanity.com/paper/1802.08760/) (2018) - [Incorporating Symmetry into Deep Dynamics Models for Improved Generalization](https://mlsanity.com/paper/2002.03061/) (2020) - [Automatic Symmetry Discovery with Lie Algebra Convolutional Network](https://mlsanity.com/paper/2109.07103/) (2021) - [Lie Point Symmetry Data Augmentation for Neural PDE Solvers](https://mlsanity.com/paper/2202.07643/) (2022) - [The Lie Derivative for Measuring Learned Equivariance](https://mlsanity.com/paper/2210.02984/) (2022) - [Self-supervised learning of Split Invariant Equivariant representations](https://mlsanity.com/paper/2302.10283/) (2023) - [Generative Adversarial Symmetry Discovery](https://mlsanity.com/paper/2302.00236/) (2023) - [3DLinker: An E(3) Equivariant Variational Autoencoder for Molecular Linker Design](https://mlsanity.com/paper/2205.07309/) (2022) - [Machine-learning hidden symmetries](https://mlsanity.com/paper/2109.09721/) (2021) - [Self-Supervised Learning with Lie Symmetries for Partial Differential Equations](https://mlsanity.com/paper/2307.05432/) (2023) - [Latent Space Symmetry Discovery](https://mlsanity.com/paper/2310.00105/) (2023) - [Deep Learning Symmetries and Their Lie Groups, Algebras, and Subalgebras from First Principles](https://mlsanity.com/paper/2301.05638/) (2023) - [Group Equivariant Generative Adversarial Networks](https://mlsanity.com/paper/2005.01683/) (2020) ## Built on by (most cited first, 17 of 17) - [Think like a Scientist: Physics-guided LLM Agent for Equation Discovery](https://mlsanity.com/paper/2602.12259/) (2026) - [Discovering Symbolic Differential Equations with Symmetry Invariants](https://mlsanity.com/paper/2505.12083/) (2025) - [Explicit Discovery of Nonlinear Symmetries from Dynamic Data](https://mlsanity.com/paper/2510.01855/) (2025) - [Governing Equation Discovery from Data Based on Differential Invariants](https://mlsanity.com/paper/2505.18798/) (2025) - [Discovering Symmetry Groups with Flow Matching](https://mlsanity.com/paper/2512.20043/) (2025) - [Adaptive Symmetry Discovery for Dynamical System Identification](https://mlsanity.com/paper/2608.08091/) (2026) - [Efficient Learning and Symmetry Discovery under Exact Invariances](https://mlsanity.com/paper/2609.07031/) (2026) - [Learning finite symmetry groups of dynamical systems via equivariance detection](https://mlsanity.com/paper/2503.03014/) (2025) - [Learning Equivariant Functions via Quadratic Forms](https://mlsanity.com/paper/2509.22184/) (2025) - [RECON: Robust symmetry discovery via Explicit Canonical Orientation Normalization](https://mlsanity.com/paper/2505.13289/) (2025) - [Recovering Physical Dynamics from Discrete Observations via Intrinsic Differential Consistency](https://mlsanity.com/paper/2605.08454/) (2026) - [Interpretable Discovery of One-parameter Subgroups: A Modular Framework for Elliptical, Hyperbolic, and Parabolic Symmetries](https://mlsanity.com/paper/2509.22219/) (2025) - [Spectral Discovery of Continuous Symmetries via Generalized Fourier Transforms](https://mlsanity.com/paper/2603.07299/) (2026) - [EqOD: Symmetry-Informed Stability Selection for PDE Identification](https://mlsanity.com/paper/2605.11524/) (2026) - [LieStoNet: Learning Lie Symmetries from Spatiotemporal Data for Stochastic Dynamical Systems](https://mlsanity.com/paper/2608.01582/) (2026) - [Discovering Symmetries in Neural Network Parameter Spaces](https://mlsanity.com/paper/2609.33527/) (2026) - [LieDiscover: Adaptive Symbolic Library Construction for Explicit Open-form Symmetry Discovery](https://mlsanity.com/paper/2609.33663/) (2026) ## Abstract (arXiv) Exploiting symmetry inherent in data can significantly improve the sample efficiency of a learning procedure and the generalization of learned models. When data clearly reveals underlying symmetry, leveraging this symmetry can naturally inform the design of model architectures or learning strategies. Yet, in numerous real-world scenarios, identifying the specific symmetry within a given data distribution often proves ambiguous. To tackle this, some existing works learn symmetry in a data-driven manner, parameterizing and learning expected symmetry through data. However, these methods often rely on explicit knowledge, such as pre-defined Lie groups, which are typically restricted to linear or affine transformations. In this paper, we propose a novel symmetry learning algorithm based on transformations defined with one-parameter groups, continuously parameterized transformations flowing along the directions of vector fields called infinitesimal generators. Our method is built upon minimal inductive biases, encompassing not only commonly utilized symmetries rooted in Lie groups but also extending to symmetries derived from nonlinear generators. To learn these symmetries, we introduce a notion of a validity score that examine whether the transformed data is still valid for the given task. The validity score is designed to be fully differentiable and easily computable, enabling effective searches for transformations that achieve symmetries innate to the data. We apply our method mainly in two domains: image data and partial differential equations, and demonstrate its advantages. Our codes are available at urlhttps://github.com/kogyeonghoon/learning-symmetry-from-scratch.git.