从轨迹数据中自动识别动力系统的对称群，无需知道方程。
Learning finite symmetry groups of dynamical systems via equivariance detection
- 通过优化等变性与冗余惩罚的损失函数发现对称变换。
- 在已知和未知方程场景下均能准确识别对称群。
- 适用于无方程先验的动力系统分析，适合物理建模研究者。
本文提出等变性探测模型（ESM），一种数据驱动方法，用于发现任意函数的有限等变对称群。ESM通过优化一个平衡等变性保持与冗余解惩罚的损失函数，确保所有对称变换的完整且准确识别。该框架专门应用于动力系统，直接从观测轨迹数据中识别其对称群。为验证其通用性，我们在两种不同场景下测试ESM：(i) 当系统控制方程理论上已知时；(ii) 当方程未知，仅依赖观测数据进行等变性发现。后一情形凸显了ESM完全数据驱动的能力，无需系统方程的先验知识即可运行。
原文摘要 · Abstract (English)
In this work, we introduce the Equivariance Seeker Model (ESM), a data-driven method for discovering the underlying finite equivariant symmetry group of an arbitrary function. ESM achieves this by optimizing a loss function that balances equivariance preservation with the penalization of redundant solutions, ensuring the complete and accurate identification of all symmetry transformations. We apply this framework specifically to dynamical systems, identifying their symmetry groups directly from observed trajectory data. To demonstrate its versatility, we test ESM on multiple systems in two distinct scenarios: (i) when the governing equations are known theoretically and (ii) when they are unknown, and the equivariance finding relies solely on observed data. The latter case highlights ESM's fully data-driven capability, as it requires no prior knowledge of the system's equations to operate.
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