Attractor Geometry Determines the Identifiability Limits of System Discovery

This paper investigates why certain dynamical regimes (fixed points, limit cycles, or chaos) are easier to recover than others when using data-driven symbolic discovery methods. The authors seek to determine if there is a fundamental, algorithm-independent limit to system identifiability that can be calculated from trajectory data before running any discovery algorithm.

Using the Lorenz-84 atmospheric model, the authors perform a controlled experiment where a single forcing parameter drives the system through various dynamical regimes while keeping the governing equations and algorithm search spaces constant. They introduce a new metric, the smallest eigenvalue of the invariant-measure moment matrix, denoted as λmin(M), to measure how well the attractor covers the function space. They evaluate two prominent discovery methods—SINDy (sparse regression) and PySR (evolutionary symbolic regression)—across varying data volumes, noise levels, and structural priors, using a newly introduced "Soft F1" score to measure structural recovery accuracy.

• Attractor Coverage as a Ceiling: The value of λmin(M) acts as a universal predictor for recovery success. When this value vanishes, recovery is impossible for any algorithm; as it increases, the potential for successful discovery improves.
• Regime Ordering: Recovery is consistently hardest at fixed points, intermediate on limit cycles, and easiest in chaotic regimes. This ordering is robust across different algorithms.
• The Chaos Paradox: While chaos generally improves identifiability by spreading the attractor, it also amplifies noise. Because SINDy and PySR process noise differently (linearly vs. superlinearly), excessive chaos can sometimes degrade performance, meaning "more chaos" is not always better.
• Mechanistic Transferability: The framework successfully predicts performance on a held-out Lorenz-96 system without any parameter refitting, confirming that the identified limits are rooted in the underlying mechanics of the dynamical system rather than algorithm-specific artifacts.