Deficiency of equation-finding approach to data-driven modeling of dynamical systems

2025-09-03

AI Summary

This work exposes a fundamental limitation of conventional sparse optimization-based equation discovery for modeling chaotic systems: equations inferred from different measurements—though capable of generating highly similar chaotic attractors—lack uniqueness and physical interpretability, leading to potentially misleading inferences. Integrating sparse regression, Koopman spectral analysis, and numerical simulations, the study systematically examines multiple chaotic systems and reveals that small-magnitude Koopman eigenvalues are highly sensitive to measurement perturbations, whereas only large-magnitude eigenvalues remain robust. This undermines the prevailing “unique correct equation” paradigm. The key contribution is the first operator-spectral proof that the deterministic assumption underlying equation discovery fails for chaotic dynamics. Consequently, the paper argues that for strongly nonlinear, highly sensitive systems, end-to-end data-driven modeling—particularly machine learning approaches—should supersede the pursuit of explicit differential equations.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Machine learning and data science for the Web
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems

Abstract

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.

Problem

• Sparse optimization methods produce sensitive equation models from imperfect data
• Different equation sets generate identical chaotic attractors despite structural differences
• Governing equation discovery may mislead physical interpretation of dynamical systems

Innovation

• Sparse optimization for equation discovery from data
• Koopman spectra analysis to compare equation sets
• Machine learning methods for direct data utilization