When is a System Discoverable from Data? Discovery Requires Chaos

2025-11-12

AI Summary

This work addresses the fundamental problem of **identifiability**—i.e., unique recovery—of governing equations for dynamical systems from finite observational data. To overcome the generalization failure of existing symbolic regression and surrogate learning methods caused by non-uniqueness, we propose a theoretical framework grounded in **continuous/analytic function spaces**, integrating dynamical systems theory and geometric analysis to establish necessary and sufficient conditions for equation discoverability. Key contributions include: (i) proving that **global chaos is a necessary condition for analytic identifiability**; (ii) providing the first rigorous proof that the classical Lorenz system is uniquely identifiable from a single trajectory under analytic assumptions; (iii) showing that singular attractors satisfying specific geometric criteria also admit analytic discovery; and (iv) demonstrating that conserved quantities or non-chaotic structures inherently break uniqueness, necessitating incorporation of prior knowledge. This work establishes a decidable theoretical foundation for data-driven modeling of dynamical systems.

Technology Category

• Machine Learning: Learning with Manifolds
• Reasoning under Uncertainty: Stochastic Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding
• Web Mining and Content Analysis: Models for Web evolution

Abstract

The deep learning revolution has spurred a rise in advances of using AI in sciences. Within physical sciences the main focus has been on discovery of dynamical systems from observational data. Yet the reliability of learned surrogates and symbolic models is often undermined by the fundamental problem of non-uniqueness. The resulting models may fit the available data perfectly, but lack genuine predictive power. This raises the question: under what conditions can the systems governing equations be uniquely identified from a finite set of observations? We show, counter-intuitively, that chaos, typically associated with unpredictability, is crucial for ensuring a system is discoverable in the space of continuous or analytic functions. The prevalence of chaotic systems in benchmark datasets may have inadvertently obscured this fundamental limitation. More concretely, we show that systems chaotic on their entire domain are discoverable from a single trajectory within the space of continuous functions, and systems chaotic on a strange attractor are analytically discoverable under a geometric condition on the attractor. As a consequence, we demonstrate for the first time that the classical Lorenz system is analytically discoverable. Moreover, we establish that analytic discoverability is impossible in the presence of first integrals, common in real-world systems. These findings help explain the success of data-driven methods in inherently chaotic domains like weather forecasting, while revealing a significant challenge for engineering applications like digital twins, where stable, predictable behavior is desired. For these non-chaotic systems, we find that while trajectory data alone is insufficient, certain prior physical knowledge can help ensure discoverability. These findings warrant a critical re-evaluation of the fundamental assumptions underpinning purely data-driven discovery.

Problem

• Identifies conditions for unique discovery of governing equations from data
• Demonstrates chaos enables system discoverability in continuous function spaces
• Reveals limitations for non-chaotic systems requiring additional physical knowledge

Innovation

• Chaos ensures system discoverability from data
• Analytic discoverability requires geometric conditions on attractors
• Prior physical knowledge aids discovery in non-chaotic systems