Zubov-Koopman Learning of Maximal Lyapunov Functions
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Bibliographic record
Abstract
While there has been increasing interest in solving Zubov's equation to find the maximal Lyapunov function, it remains a challenge for dynamical systems with limited knowledge of system dynamics. In this paper, we present a Zubov-Koopman approach to learning a Lyapunov function that is nearly maximal for an unknown nonlinear system but has a known equilibrium point. The proposed approach is a lifting approach to map observable data into an infinite-dimensional function space, which generates a flow governed by our proposed ‘Zubov-Koopman’ operator. By learning a Zubov-Koopman operator over a fixed time interval, we can indirectly approximate the solution to Zubov's Equation through iterative application of the learned operator on the identity function. We also demonstrate that a transformation of such an approximator can be readily utilized as a near-maximal Lyapunov function. We present an algorithm for learning Zubov-Koopman operators, asserting that this method not only decreases the necessary data volume but also achieves favorable outcomes in estimating regions of attraction, as illustrated by numerical examples.
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Full frame distilled prediction
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) | 0.006 | 0.000 |
Machine scores (provisional)
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Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it