  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.
