Out-of-equilibrium dynamical mean-field equations for the perceptron model

TL;DR

This paper derives general out-of-equilibrium dynamical mean-field equations for the perceptron model, enabling analysis of complex systems like neural networks, glasses, and ecosystems under arbitrary noise and friction conditions.

Contribution

It introduces a novel derivation of mean-field dynamical equations for the perceptron using cavity and path-integral methods, applicable to non-equilibrium scenarios.

Findings

Derived general dynamical equations for perceptrons with arbitrary noise and friction.

Reduced complex dynamics to an effective stochastic process for a representative variable.

Potential applications include studying glass transitions, jamming, and rheology in high-dimensional systems.

Abstract

Perceptrons are the building blocks of many theoretical approaches to a wide range of complex systems, ranging from neural networks and deep learning machines, to constraint satisfaction problems, glasses and ecosystems. Despite their applicability and importance, a detailed study of their Langevin dynamics has never been performed yet. Here we derive the mean-field dynamical equations that describe the continuous random perceptron in the thermodynamic limit, in a very general setting with arbitrary noise and friction kernels, not necessarily related by equilibrium relations. We derive the equations in two ways: via a dynamical cavity method, and via a path-integral approach in its supersymmetric formulation. The end point of both approaches is the reduction of the dynamics of the system to an effective stochastic process for a representative dynamical variable. Because the perceptron is formally very close to a system of interacting particles in a high dimensional space, the methods we develop here can be transferred to the study of liquid and glasses in high dimensions. Potentially interesting applications are thus the study of the glass transition in active matter, the study of the dynamics around the jamming transition, and the calculation of rheological properties in driven systems.