Information theory and discriminative sampling for model discovery
This work addresses the challenge of reducing data requirements for researchers in dynamical systems modeling, though it is incremental as it applies existing information theory tools to an established sparse identification framework.
The authors tackled the problem of improving sampling efficiency and model performance in data-driven discovery of nonlinear dynamics by leveraging Fisher information and Shannon entropy to prioritize informative data. They demonstrated enhanced data efficiency in chaotic and non-chaotic systems, including scenarios with single trajectories, tunable parameters, and multiple initializations.
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.