Nearest-Neighbours Neural Network architecture for efficient sampling of statistical physics models

TL;DR

This paper introduces the 4N neural network architecture, which efficiently learns the Gibbs-Boltzmann distribution in disordered systems like spin glasses, improving sampling accuracy and interpretability.

Contribution

The 4N architecture is a physically-interpretable neural network with linear parameter scaling, capable of accurately modeling complex disordered systems and linking performance to correlation length.

Findings

4N accurately learns the Gibbs-Boltzmann distribution for 2D Edwards-Anderson model.

Performance improves with more layers, correlating with system's correlation length.

The architecture is applicable to various topologies and captures key physical properties.

Abstract

The task of sampling efficiently the Gibbs-Boltzmann distribution of disordered systems is important both for the theoretical understanding of these models and for the solution of practical optimization problems. Unfortunately, this task is known to be hard, especially for spin glasses at low temperatures. Recently, many attempts have been made to tackle the problem by mixing classical Monte Carlo schemes with newly devised Neural Networks that learn to propose smart moves. In this article we introduce the Nearest-Neighbours Neural Network (4N) architecture, a physically-interpretable deep architecture whose number of parameters scales linearly with the size of the system and that can be applied to a large variety of topologies. We show that the 4N architecture can accurately learn the Gibbs-Boltzmann distribution for the two-dimensional Edwards-Anderson model, and specifically for some of its most difficult instances. In particular, it captures properties such as the energy, the correlation function and the overlap probability distribution. Finally, we show that the 4N performance increases with the number of layers, in a way that clearly connects to the correlation length of the system, thus providing a simple and interpretable criterion to choose the optimal depth.