Nearest-neighbors neural network architecture for efficient sampling of statistical physics models

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-glass-like problems 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-neighbors neural network ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mtext>4N</mml:mtext> </mml:mrow> </mml:math> ) 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mtext>4N</mml:mtext> </mml:mrow> </mml:math> architecture can accurately learn the Gibbs–Boltzmann distribution for a prototypical spin-glass model, 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mtext>4N</mml:mtext> </mml:mrow> </mml:math> 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.