Fisher information and Shannon entropy are fundamental tools for understanding and analyzing dynamical systems from complementary perspectives. They can characterize unknown parameters by quantifying the information contained in variables, or measure how different initial trajectories or temporal segments of a trajectory contribute to learning or inferring system dynamics. In this work, we leverage the Fisher Information Matrix (FIM) within the data-driven framework of {\em sparse identification of nonlinear dynamics} (SINDy). We visualize information patterns in chaotic and non-chaotic systems for both single trajectories and multiple initial conditions, demonstrating how information-based analysis can improve sampling efficiency and enhance model performance by prioritizing more informative data. The benefits of statistical bagging are further elucidated through spectral analysis of the FIM. We also illustrate how Fisher information and entropy metrics can promote data efficiency in three scenarios: when only a single trajectory is available, when a tunable control parameter exists, and when multiple trajectories can be freely initialized. As data-driven model discovery continues to gain prominence, principled sampling strategies guided by quantifiable information metrics offer a powerful approach for improving learning efficiency and reducing data requirements.

翻译：费希尔信息与香农熵是从互补视角理解和分析动力系统的基本工具。它们可通过量化变量中包含的信息来表征未知参数，或衡量不同初始轨迹或轨迹的时间段对学习或推断系统动力学的贡献。在本工作中，我们将费希尔信息矩阵（FIM）应用于数据驱动的非线性动力学稀疏识别（SINDy）框架中。我们可视化了混沌与非混沌系统中单条轨迹及多初始条件的信息模式，证明了基于信息的分析如何通过优先选择信息量更大的数据来提高采样效率并增强模型性能。通过对FIM的谱分析进一步阐明了统计袋装法的优势。我们还展示了费希尔信息与熵度量在三种场景中如何提升数据效率：仅有一条轨迹可用时、存在可调控制参数时，以及可自由初始化多条轨迹时。随着数据驱动的模型发现日益受到重视，由可量化信息指标指导的原则性采样策略为提升学习效率、降低数据需求提供了有力途径。