Data-Driven Domain of Attraction Estimation via Convergent Koopman-Zubov Approximation
arXiv:2608.01018
2026
Dynamics
2 ideas extracted · analyzed Aug 31, 2026
What the math gives to ML
The paper provides a constructive way to estimate a nonlinear system's domain of attraction by converting the Zubov function into a unique invariant element of a Koopman-like operator acting on an RKHS. Its key transferable mechanism is spectral contraction: a linear-radial product kernel makes the infinite-time operator have spectrum strictly inside the unit circle, so repeated operator application converges and estimation error decreases with sample size. This can be transferred to recurrent, neural-ODE, and world-model networks as an auxiliary Lyapunov or attraction certificate. The strongest experiments measure invariance residuals, spectral contraction, and the boundary between stable and unstable long-horizon rollouts.
Ideas from this paper
△ Mechanism confirmed, baseline not beaten
2026
Attach a scalar Zubov head to a neural ODE, state-space model, or recurrent world model and train it to be invariant under a discounted Koopman action. The head should be near one for trajectories attracted to the target equilibrium and near zero for states with large accumulated deviation, providing a long-horizon stability signal and an off-distribution failure detector.
Useful8/10
Difficulty6/10
Novelty7/10
✗ Mechanism failed
2026
Use a convergent kernel approximation of the Zubov invariant as a trust-region monitor for a learned dynamics model. The estimated Zubov sublevel sets become an inference-time gate that rejects, shortens, or dampens transitions predicted to leave the learned attraction region.
Useful7/10
Difficulty5/10
Novelty8/10