Symmetries and phase diagrams with real-space mutual information neural estimation

TL;DR

This paper introduces RSMI-NE, an unsupervised neural network algorithm that efficiently estimates real-space mutual information to identify relevant degrees of freedom, symmetries, and phase diagrams in physical systems from configurations.

Contribution

It combines variational mutual information bounds with differentiable coarse-graining to develop a novel neural network method for extracting effective field theory operators directly from data.

Findings

Successfully extracts phase diagrams and order parameters.

Identifies emergent and explicit symmetries.

Applies to both equilibrium and non-equilibrium systems.

Abstract

Real-space mutual information (RSMI) was shown to be an important quantity, formally and from a numerical standpoint, in finding coarse-grained descriptions of physical systems. It very generally quantifies spatial correlations, and can give rise to constructive algorithms extracting relevant degrees of freedom. Efficient and reliable estimation or maximization of RSMI is, however, numerically challenging. A recent breakthrough in theoretical machine learning has been the introduction of variational lower bounds for mutual information, parametrized by neural networks. Here we describe in detail how these results can be combined with differentiable coarse-graining operations to develop a single unsupervised neural-network based algorithm, the RSMI-NE, efficiently extracting the relevant degrees of freedom in the form of the operators of effective field theories, directly from real-space configurations. We study the information contained in the statistical ensemble of constructed coarse-graining transformations, and its recovery from partial input data using a secondary machine learning analysis applied to this ensemble. In particular, we show how symmetries, also emergent, can be identified. We demonstrate the extraction of the phase diagram and the order parameters for equilibrium systems, and consider also an example of a non-equilibrium problem.