发表机构
Harvard University; School of Engineering and Applied Sciences (SEAS), Harvard University; OEB, Harvard University（哈佛大学; 哈佛大学工程与应用科学学院; 哈佛大学有机与进化生物学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
LieStoNet是端到端无模板框架，可从时空轨迹发现SDE的李点对称性，在已知解析对称的典型SDE上恢复出与真实对称代数一致的生成元，为含噪动力学提供可解释对称性发现方案。
AI 中文摘要
对称性是现代机器学习与物理学的核心：不变性和等变性可提升样本效率、鲁棒性及分布外泛化能力，而对称原理为科学建模提供指导。然而对于随机动力系统，相关的连续对称性往往未知，随机微分方程（SDE）的对称性发现问题基本未被探索。我们提出LieStoNet，这是一个端到端、无模板的框架，可直接从时空轨迹中发现SDE的李点对称性，无需预先指定对称群、模板或正则坐标。该框架基于Gaeta和Quintero（1999）提出的开创性SDE李对称理论，该理论将SDE的李点对称性及其与福克-普朗克（Fokker-Planck）对称性的关系形式化；LieStoNet从增量中学习漂移项和扩散项的神经代理，随后通过满足SDE确定方程学习可投影生成元，分别对李括号下的封闭性、李代数公理（双线性、反对称性、雅可比恒等式）的遵守情况以及非冗余独立基进行正则化。该代理还定义了相关的福克-普朗克方程，支持并行发现其李点对称性。在多个具有已知解析对称性的典型SDE上，LieStoNet恢复出与真实对称代数一致的生成元，为含噪动力学提供可解释的对称性发现方案。代码可在https://github.com/...（注：原链接为this https URL，此处保留原表述）获取。
英文摘要
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}.
Comments25 Pages, 7 figures. Accepted to the International Conference on Machine Learning (ICML 2026)
