LieStoNet: Learning Lie Symmetries from Spatiotemporal Data for Stochastic Dynamical Systems

2026-08-02

AI Summary

This work addresses the challenge that Lie point symmetries of stochastic differential equations (SDEs) are typically unknown and lack automated discovery methods from data. The authors propose LieStoNet, a novel framework that, for the first time, enables end-to-end learning of SDE Lie point symmetries directly from spatiotemporal trajectories without requiring predefined symmetry groups or templates. Built upon the SDE symmetry theory of Gaeta and Quintero, LieStoNet models drift and diffusion terms via neural networks and jointly uncovers symmetries of the associated Fokker–Planck equation. To guarantee that the learned infinitesimal generators form a valid Lie algebra, the method incorporates deterministic equation constraints, Lie bracket closure, and algebraic regularization. Evaluated on several canonical SDEs with known analytical symmetries, LieStoNet accurately recovers the symmetry generators, demonstrating its capability for interpretable symmetry discovery in noisy dynamical systems.

Abstract

Symmetry is central to modern machine learning and physics: invariances and equivariances improve sample efficiency, robustness, and out-of-distribution generalization, while symmetry principles guide scientific modeling. Yet for stochastic dynamical systems the relevant continuous symmetries are rarely known, and symmetry discovery for SDEs has remained essentially unexplored. We introduce \textit{LieStoNet}, an end-to-end, \emph{template-free} framework for discovering Lie-point symmetries of SDEs directly from spatiotemporal trajectories, without prespecifying symmetry groups, templates, or canonical coordinates. Building on the seminal SDE Lie-symmetry theory of Gaeta and Quintero (1999), which formalizes Lie-point SDE symmetries and their relation to Fokker-Planck symmetries, LieStoNet learns neural surrogates for drift and diffusion from increments, then learns projectable generators by enforcing the SDE determining equations, separately regularizing for closure under Lie brackets, adherence to the Lie algebra axioms (bilinearity, antisymmetry, Jacobi), and a non-redundant independent basis. The surrogate also defines an associated Fokker-Planck equation, enabling optional discovery of its Lie-point symmetries in parallel. Across multiple canonical SDEs with known analytic symmetries, LieStoNet recovers generators consistent with the ground-truth symmetry algebra, providing interpretable symmetry discovery for noisy dynamics. Code is available at \href{https://github.com/sumit-sinha-seas/LieStoNet_Final.git}{this link}.

Problem

• stochastic dynamical systems
• Lie symmetries
• symmetry discovery
• SDEs
• spatiotemporal data

Innovation

• Lie symmetry
• stochastic differential equations
• neural surrogate
• Fokker-Planck equation
• symmetry discovery