LieStoNet: Learning Lie Symmetries from Spatiotemporal Data for Stochastic Dynamical Systems
2026-08-03 • Machine Learning
Machine Learning
AI summaryⓘ
The authors introduce LieStoNet, a new method that finds hidden continuous symmetries in stochastic differential equations (SDEs) using only data from system behavior, without needing prior knowledge about the symmetries. Their approach learns how the system changes over time and identifies symmetry generators by following mathematical rules from SDE theory. They tested LieStoNet on several well-known SDEs and showed it can accurately recover known symmetries, even from noisy data. This work helps understand complex random systems by revealing their underlying structure.
SymmetryStochastic Differential EquationsLie-point SymmetriesFokker-Planck EquationDriftDiffusionLie AlgebraNeural SurrogatesEquivarianceStochastic Dynamical Systems
Authors
Shida Liu, Abhishek Gupta, Sumit Sinha, L. Mahadevan
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}.