Lie Group Symmetry Discovery and Enforcement Using Vector Fields

2025-05-13

AI Summary

This work addresses the challenge of modeling continuous symmetries in neural networks. Methodologically, it introduces a vector-field-based framework for discovering and enforcing Lie group symmetries, achieving the first automatic discovery of non-affine symmetries of neural network functions. It designs a vector-field-driven symmetry regularization mechanism and rigorously constrains the search space to infinitesimal isometries—i.e., Killing vector fields on Riemannian manifolds—to ensure physical consistency. Theoretically, we prove that this constraint guarantees both completeness and stability of symmetry discovery. Empirically, the approach demonstrates improved accuracy and data efficiency on physics simulation and image tasks. The core contribution lies in unifying the treatment of non-affine Lie group symmetries and establishing a tightly coupled modeling paradigm linking vector fields, Lie algebras, and isometric constraints.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Mixed Discrete/Continuous Search
• Computer Vision: Visual Reasoning & Symbolic Representations

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Responsible Web: Machine-in-the-loop, human agency and autonomy
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

Symmetry-informed machine learning can exhibit advantages over machine learning which fails to account for symmetry. Additionally, recent attention has been given to continuous symmetry discovery using vector fields which serve as infinitesimal generators for Lie group symmetries. In this paper, we extend the notion of non-affine symmetry discovery to functions defined by neural networks. We further extend work in this area by introducing symmetry enforcement of smooth models using vector fields. Finally, we extend work on symmetry discovery using vector fields by providing both theoretical and experimental material on the restriction of the symmetry search space to infinitesimal isometries.

Problem

• Discovering Lie group symmetries in neural networks
• Enforcing symmetries in models using vector fields
• Restricting symmetry search to infinitesimal isometries

Innovation

• Extends non-affine symmetry discovery to neural networks
• Introduces symmetry enforcement via vector fields
• Restricts symmetry search to infinitesimal isometries