Global minimization via classical tunneling assisted by collective force field formation
This mechanism offers a new approach to global minimization for systems with exponentially growing equilibrium points, potentially benefiting optimization, Monte Carlo schemes, and machine learning.
This paper describes a phenomenon where increasing dimensions in simple dynamical models generates a self-consistent force field due to dynamical instabilities. This "Lyapunov force" steers the system towards the global minimum of a potential function, even when the number of equilibrium points grows exponentially with system size.
Simple dynamical models can produce intricate behaviors in large networks. These behaviors can often be observed in a wide variety of physical systems captured by the network of interactions. Here we describe a phenomenon where the increase of dimensions self-consistently generates a force field due to dynamical instabilities. This can be understood as an unstable ("rumbling") tunneling mechanism between minima in an effective potential. We dub this collective and nonperturbative effect a "Lyapunov force" which steers the system towards the global minimum of the potential function, even if the full system has a constellation of equilibrium points growing exponentially with the system size. The system we study has a simple mapping to a flow network, equivalent to current-driven memristors. The mechanism is appealing for its physical relevance in nanoscale physics, and to possible applications in optimization, novel Monte Carlo schemes and machine learning.