Phase diagrams with real-space mutual information neural estimation

Real-space mutual information (RSMI) has been shown to be an important quantity, both formally and from numerical standpoint, in constructing 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, e.g. about the symmetries, contained in the ensemble of constructed coarse-graining transformations, and its practical recovery from partial input data using a secondary machine learning analysis applied to this ensemble. We demonstrate how the phase diagram and the order parameters for equilibrium systems are extracted, and consider also an example of a non-equilibrium problem.