A Green Function Approach to Smooth Nonautonomous Topological Equivalence with Unbounded Nonlinearities under $(μ,ν)$--Dichotomies
arXiv:2608.14715
2026
Dynamics
1 ideas extracted · analyzed Sep 1, 2026
What the math gives to ML
The paper provides a constructive Green-operator mechanism for proving that a nonautonomous linear flow remains topologically, and under stronger variational conditions smoothly, equivalent to a quasilinear perturbation. The transferable asset is a computable dichotomy margin: the perturbation is safe when its Lipschitz gain multiplied by the induced Green-operator norm is below one, while smooth invertibility requires an analogous first-derivative margin. This can be implemented as a residual-network or state-space architecture whose layer dynamics are designed with stable and unstable subspaces and whose residual gain is constrained using an estimated Green norm, yielding a falsifiable depth- and gain-dependent divergence boundary.
Ideas from this paper
Unverified
2026
Replace unconstrained residual blocks by a nonautonomous linear backbone plus a learned nonlinear perturbation, and constrain the perturbation gain using the Green operator of the backbone. The resulting network can contain both contracting and expanding channels, but the accumulated response of the perturbation remains bounded when its Green margin is below one. A differentiable or periodically updated estimate of this margin becomes both an architecture constraint and a training monitor.
Useful7/10
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