arXiv:2609.02458 2026 Dynamics 2 ideas extracted · analyzed Sep 3, 2026
The paper gives a constructive linear-response description of periodically driven finite-state stochastic dynamics: each current response can be represented by an equivalent circuit with N resistor-capacitor series branches. The transferable mechanism is a frequency-dependent susceptibility whose poles encode relaxation times, with low-frequency quasi-static behavior and high-frequency attenuation. In neural-network training, this can become online system identification: periodically perturb the learning rate or update magnitude, estimate the optimizer-to-loss transfer function, and use its poles to select damping and step sizes before oscillatory or unstable modes dominate.
Treat local neural-network training as a driven linear system and periodically modulate the learning rate by a small sinusoid. Estimate the transfer function from this modulation to loss or gradient observables, fit its relaxation poles, and set the learning rate below the measured instability boundary.
Use the paper's parallel-branch response structure as a measurable regularizer on training dynamics. Penalize large high-frequency gain and excessively slow relaxation modes, encouraging parameter updates whose loss response is smooth, damped, and composed of controlled time scales rather than a sharp unstable mode.