Attractor Geometry Limits System Discovery in Chaotic Dynamics.
Summary
This research shows that the "attractor geometry" of a system, measured by the smallest eigenvalue of its invariant-measure moment matrix, sets a fundamental limit on how well governing equations can be discovered from data. It demonstrates that this metric predicts the identifiability ceiling for both sparse regression and evolutionary symbolic regression algorithms.
Why it matters
Professionals developing or applying AI for system identification and modeling need to understand these inherent limitations to set realistic expectations and choose appropriate methodologies. This work provides a theoretical framework to predict when system discovery will be feasible and how different algorithms might perform.
How to implement this in your domain
- 1Evaluate the $\lambda_{\min}(M)$ metric for your system's data before attempting symbolic regression to assess identifiability.
- 2Consider the impact of system chaos and noise amplification when selecting between sparse regression (SINDy) and evolutionary symbolic regression (PySR).
- 3Integrate the Soft F1 metric into your evaluation pipeline for a more nuanced assessment of model structural accuracy.
- 4Prioritize data collection strategies that maximize attractor coverage in function space to improve system discovery potential.
Who benefits
Key takeaways
- The geometry of a system's attractor fundamentally limits the identifiability of its governing equations.
- A new metric, $\lambda_{\min}(M)$, can predict the ceiling for system discovery algorithms.
- Chaos can improve identifiability by spreading the attractor but also amplifies noise, affecting algorithms differently.
- A new Soft F1 metric offers a more precise way to evaluate structural performance in symbolic regression.
Original post by Matteo Gallo, Fabio Anselmi, Paolo Lazzari
"arXiv:2607.18490v1 Announce Type: new Abstract: Symbolic discovery of governing equations from data is limited not only by algorithm design and data volume, but by the geometry of the attractor: what the long-run dynamics allow to be recovered. Using a within-system design on Lor…"
View on XOriginally posted by Matteo Gallo, Fabio Anselmi, Paolo Lazzari on X · view source
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