From Flows to Maps: Sampling Laws for Attractor Intensity and Bounded-Noise Escape
arXiv:2608.02933
2026
Dynamics
2 ideas extracted · analyzed Aug 31, 2026
What the math gives to ML
The paper gives a constructive sampling law for the largest persistent bounded disturbance that an attracting dynamical system can tolerate while remaining confined to its basin. For an L-Lipschitz vector field, the normalized discrete intensity of an exact time-h map converges to the continuous-time intensity with explicit two-sided bounds and generally sharp O(h) error. This mechanism transfers to neural optimizers, recurrent state-space models, and iterative inference procedures by separating per-step perturbation budgets from disturbance rates per unit computation time. A practical implementation can estimate an empirical escape threshold across step sizes and use the sampling law as a stability certificate or adaptive step-size controller.
Ideas from this paper
△ Mechanism confirmed, baseline not beaten
2026
Treat optimizer or recurrent-network updates as sampled observations of an underlying continuous-time flow, and measure robustness using disturbance amplitude divided by the sampling interval. Estimate the largest persistent perturbation that keeps trajectories inside a chosen attracting basin, then transfer this estimate across learning rates or inference step sizes using the paper's explicit sampling bounds.
Useful8/10
Difficulty5/10
Novelty8/10
✓✓ Beats tuned baseline
2026
Use bounded-noise escape as a measurable stability transition to adapt the learning rate or recurrent integration step before catastrophic loss of confinement. Periodically estimate the disturbance radius at which the current training dynamics exits its stable region, then adjust the step size to maintain a fixed safety margin.
Useful7/10
Difficulty6/10
Novelty9/10