数据驱动的方程发现法在混沌系统中不可靠，不同方程却产生相同混沌吸引子。
Deficiency of equation-finding approach to data-driven modeling of dynamical systems
- 用稀疏优化从有噪声数据中找动力系统方程
- 不同方程组生成几乎相同的混沌吸引子，但小特征值差异明显
- 适合关注数据建模局限性的研究人员
通过稀疏优化从数据中寻找控制方程已成为确定性动力系统数据驱动建模的流行方法。考虑到数据因扰动和测量误差可能不完美，我们发现对于许多混沌系统，广泛使用的稀疏优化方法所发现的方程对测量过程极为敏感，但所有这些模型均产生几乎相同的混沌吸引子，这一现象对复杂动力系统中基于方程建模的传统观念提出了严峻挑战。通过计算Koopman谱，我们发现不同方程组在较大特征值上一致，差异仅在特征值低于特定阈值时显现。结果表明，试图从方程中获取物理意义可能误导结论，直接使用机器学习处理原始数据或更为有效。
原文摘要 · Abstract (English)
Finding the governing equations from data by sparse optimization has become a popular approach to deterministic modeling of dynamical systems. Considering the physical situations where the data can be imperfect due to disturbances and measurement errors, we show that for many chaotic systems, widely used sparse-optimization methods for discovering governing equations produce models that depend sensitively on the measurement procedure, yet all such models generate virtually identical chaotic attractors, leading to a striking limitation that challenges the conventional notion of equation-based modeling in complex dynamical systems. Calculating the Koopman spectra, we find that the different sets of equations agree in their large eigenvalues and the differences begin to appear when the eigenvalues are smaller than an equation-dependent threshold. The results suggest that finding the governing equations of the system and attempting to interpret them physically may lead to misleading conclusions. It would be more useful to work directly with the available data using, e.g., machine-learning methods.
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