Information theory and discriminative sampling for model discovery

TL;DR

This paper integrates Fisher information and Shannon entropy into the SINDy framework to analyze dynamical systems, demonstrating how information metrics can optimize sampling and improve model discovery efficiency.

Contribution

It introduces a novel approach combining information theory with sparse identification of nonlinear dynamics to enhance data efficiency and sampling strategies.

Findings

Information patterns differ between chaotic and non-chaotic systems.

Information-based analysis improves sampling efficiency and model accuracy.

Fisher information and entropy metrics promote data efficiency in various scenarios.

Abstract

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.