LieDynNet: Learning Lie Symmetries from Spatiotemporal Data

Shida Liu, Sumit Sinha, Vishaal Krishnan, L Mahadevan

Keywords: Lie Symmetries, Dynamical Systems, ODE/PDEs, Neural Networks, Unsupervised Learning

TL;DR: LieDynNet learns Lie symmetry generators for unknown ODEs/PDEs from data by pairing neural dynamics surrogates with IIC and Lie-algebra constraints, validated by finite flows, to recover symmetry algebras and invariants.

Abstract: Continuous symmetries of dynamical systems—transformations that map solution trajectories or spatiotemporal fields to new, valid solutions—are powerful tools for analysis, reduction, and control. Prior work on symmetry discovery broadly falls into two categories: methods that prioritize Lie-algebraic structure but operate on static datasets rather than dynamical systems, and methods that discover symmetries for dynamical systems but often do not enforce algebraic structure. Across both threads, most approaches also neglect the infinitesimal invariance condition (IIC)—that prolonged generators annihilate the governing equations. To fill this gap, we introduce LieDynNet, which learns Lie symmetry generators directly from data by pairing neural ODE/PDE models with two families of constraints: dynamical validity, enforced both via IIC (via generator prolongations) and under finite flows; and algebraic soundness, enforcing closure, antisymmetry, and the Jacobi identity so the generators form a Lie algebra. The framework is model-agnostic and applies to both ODEs and PDEs without hand-crafted priors. On canonical dynamical systems, LieDynNet recovers symmetry algebras and associated invariants from data, showing that learned symmetries can be simultaneously algebraically consistent and dynamically faithful. These results provide a practical, data-driven route to discovering the symmetry structure of complex dynamical phenomena.

Primary Area: applications to physical sciences (physics, chemistry, biology, etc.)

Submission Number: 23491