Proximal Relations in Asymptotically Commutative Non-Autonomous Dynamical Systems
arXiv:2608.16917
2026
Dynamics
1 ideas extracted · analyzed Sep 1, 2026
What the math gives to ML
The paper develops a transferable mechanism for time-varying dynamical systems: asymptotic commutativity, in which the order dependence of maps vanishes at late times, together with proximal relations defined by repeated near-collisions of trajectories. For a non-autonomous neural network or recurrent system, this suggests controlling late-time commutators between update blocks so that long-horizon behavior becomes less sensitive to small scheduling or time-order perturbations. The key falsifiable signature is decay of the empirical commutator norm with time, accompanied by improved agreement between nominal and reordered trajectories without forcing all hidden states to collapse.
Ideas from this paper
Unverified
2026
Replace a time-homogeneous recurrent update by a sequence of parameterized maps f_t, and regularize late-time pairs of updates to approximately commute: applying block f_t followed by f_r should agree with applying f_r followed by f_t. This should make long-horizon predictions robust to local time-step reorderings and schedule perturbations, while proximal statistics provide a diagnostic for whether trajectories repeatedly approach one another rather than diverging permanently.
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