  Density functional theory underlies the most successful and widely used
numerical methods for electronic structure prediction of solids. However, it
has the fundamental shortcoming that the universal density functional is
unknown. In addition, the computational result---energy and charge density
distribution of the ground state---is useful for electronic properties of
solids mostly when reduced to a band structure interpretation based on the
Kohn-Sham approach. Here, we demonstrate how machine learning algorithms can
help to free density functional theory from these limitations. We study a
theory of spinless fermions on a one-dimensional lattice. The density
functional is implicitly represented by a neural network, which predicts,
besides the ground-state energy and density distribution, density-density
correlation functions. At no point do we require a band structure
interpretation. The training data, obtained via exact diagonalization, feeds
into a learning scheme inspired by active learning, which minimizes the
computational costs for data generation. We show that the network results are
of high quantitative accuracy and, despite learning on random potentials,
capture both symmetry-breaking and topological phase transitions correctly.
