Learning Regions of Attraction in Unknown Dynamical Systems via Zubov–Koopman Lifting: Regularities and Convergence
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Bibliographic record
Abstract
The estimation for the region of attraction (ROA) of an asymptotically stable equilibrium point is crucial in the analysis of nonlinear systems. There has been a recent surge of interest in estimating the solution to Zubov's equation, whose non-trivial sub-level sets form the exact ROA. In this paper, we propose a lifting approach to map observable data into an infinite-dimensional function space, which generates a flow governed by the proposed Zubov-Koopman operators. By learning a Zubov-Koopman operator over a fixed time interval, we can indirectly approximate the solution to Zubov's equation by iteratively applying the learned operator on certain functions. We also demonstrate that a transformation of such an approximator can be readily utilized as a near-maximal Lyapunov function. We approach our goal through a comprehensive investigation of the regularities of Zubov-Koopman operators and their associated quantities. Based on these findings, we present an algorithm for learning Zubov-Koopman operators that exhibit strong convergence to the true operator. We show that this approach reduces the amount of required data and can yield desirable estimation results, as demonstrated through numerical examples.
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Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
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