发表机构
Harvard John A. Paulson School of Engineering and Applied Sciences, Harvard University（哈佛大学约翰·A·保尔森工程与应用科学学院、哈佛大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究针对未知对称群等变的动力系统，提出自适应对称性发现方法，可从更短单条轨迹辨识系统，性能与已知对称性方法相当，分析依托群表示论与凯莱图性质。
AI 中文摘要
动力系统可对由固定底层动力学生成的轨迹数据进行建模，其应用范围从生物学到物理学。尤其在科学场景中，动力系统并非通用模型，而是常展现出由物理定律施加的对称性，通过关于群作用的等变性来形式化。辨识问题关注从观测轨迹中恢复系统的参数。本研究探讨用于动力系统辨识的自适应对称性发现，解决当系统关于未知对称群等变时，如何从单条轨迹中辨识该系统的问题。为此，我们首先证明，对于已知对称性的情况，与通用设置相比，可从显著更短的单条轨迹中辨识系统，并精确刻画了这种提升。接着，我们考虑自动对称性发现场景，提出一种直接从单条轨迹学习对称群并将其融入辨识过程的方法，达到了与已知对称性情况相同的最优轨迹长度。我们的分析依赖于群表示论和凯莱图的扩张器性质，这对动力系统对称性研究可能具有独立的参考价值。
英文摘要
Dynamical systems model trajectory data generated by fixed underlying dynamics, with applications ranging from biology to physics. Especially in scientific settings, dynamical systems are not generic but often exhibit symmetries imposed by physical laws, formalized through equivariance with respect to group actions. The identification problem concerns recovering the parameters of a system from observed trajectories. In this work, we study adaptive symmetry discovery for dynamical system identification and address how a system can be identified from a single trajectory when it is equivariant with respect to an unknown symmetry group. To this end, we first show that for known symmetries, the system can be identified from a significantly shorter single trajectory than in the generic setting, and we precisely characterize this improvement. We then consider the automatic symmetry discovery setting, proposing a method to learn the symmetry group directly from a single trajectory and incorporate it into the identification procedure, achieving the same optimal trajectory length as in the known-symmetry case. Our analysis relies on tools from group representation theory and the expander properties of Cayley graphs, and may be of independent interest for the study of symmetries in dynamical systems.
Comments38 pages, 1 figure. Published at ICML 2026
Journal refInternational Conference on Machine Learning (ICML) 2026
