Localization Delocalization Transition in Diffusion with Adaptive Resetting

What the math gives to ML

The paper identifies a sharp localization threshold for diffusion with position-dependent stochastic resetting: if the reset rate scales as r(x) proportional to |x|^lambda, localization occurs for lambda greater than -2, while lambda less than -2 cannot confine the diffusive particle. The marginal inverse-square case, r(x) = r0 divided by x squared, produces power-law stationary tails and a finite-noise delocalization transition because normalizability depends on the ratio r0/D. A transferable neural-network mechanism is an adaptive stochastic reset of parameters, optimizer states, or recurrent hidden states toward a checkpoint, with reset hazard proportional to inverse squared distance; this gives a measurable stability boundary rather than an arbitrary reset schedule.