  Machine learning has made important headway in helping to improve the
treatment of quantum many-body systems. A domain of particular relevance are
correlated inhomogeneous systems. What has been missing so far is a general,
scalable deep-learning approach that would enable the rapid prediction of
spatial densities for strongly correlated systems in arbitrary potentials. In
this work, we present a straightforward scheme, where we learn to predict
densities using convolutional neural networks trained on random potentials.
While we demonstrate this approach in 1D and 2D lattice models using data from
numerical techniques like Quantum Monte Carlo, it is directly applicable as
well to training data obtained from experimental quantum simulators. We train
networks that can predict the densities of multiple observables simultaneously
and that can predict for a whole class of many-body lattice models, for
arbitrary system sizes. We show that our approach can handle well the interplay
of interference and interactions and the behaviour of models with phase
transitions in inhomogeneous situations, and we also illustrate the ability to
solve inverse problems, finding a potential for a desired density.
