Weakly Normally Hyperbolic Invariant Tori: Persistence and an Averaging Principle
arXiv:2608.00812
2026
Dynamics
1 ideas extracted · analyzed Aug 31, 2026
What the math gives to ML
The paper provides a transferable persistence mechanism: an attracting weakly normally hyperbolic invariant torus survives sufficiently small time-periodic perturbations, and averaging converts an attracting torus of an averaged autonomous system into a higher-dimensional attracting torus of the periodically forced system. The useful neural-network analogue is a phase-augmented optimizer or recurrent state-space model whose transverse contraction dominates tangential expansion and perturbation strength. This yields a concrete design rule for cyclic training or inference dynamics, together with measurable Floquet-multiplier and decay-rate predictions rather than a benchmark-only claim.
Ideas from this paper
✗ Failed on benchmark
2026
Augment an optimizer with a periodic phase and deliberately use a cyclic learning-rate or momentum forcing whose averaged dynamics have an attracting low-dimensional set. Treat the resulting periodic parameter orbit as an invariant torus and tune the schedule so transverse contraction dominates tangential sensitivity and minibatch perturbations. The goal is a robust, phase-locked training orbit that explores parameter space without losing attraction toward a useful solution manifold.
Useful7/10
Difficulty5/10
Novelty7/10