arXiv:2608.16186 2026 Dynamics 1 ideas extracted · analyzed Sep 1, 2026
The paper develops a modulation method for a multi-scale dynamical system whose raw parameters are strongly coupled and lose coercivity in the critical dimension. Its transferable asset is the replacement of naive coordinates by rescaled and modified coordinates, followed by an explicit small linear system that solves for parameter velocities while suppressing first-order cross-couplings. In neural-network training, this suggests a blockwise normal-form optimizer: estimate tangent-space coupling between parameter groups, solve for preconditioned updates in rescaled coordinates, and test whether the transformed dynamics reduce oscillation and improve conditioning. The PDE-specific bubble construction is not transferable, so experiments should isolate the coordinate-transformation and modulation mechanism.
Replace raw updates of strongly coupled parameter blocks by updates in rescaled, approximately normal-form coordinates. The optimizer estimates the local coupling matrix between block directions, solves a small modulation system for transformed velocities, and optionally subtracts predictable first-order cross-block drift.