  A remarkable consequence of the Hohenberg-Kohn theorem of density functional
theory is the existence of an injective map between the electronic density and
any observable of the many electron problem in an external potential. In this
work, we study the problem of predicting a particular observable, the band gap
of semiconductors and band insulators, from the knowledge of the local
electronic density. Using state-of-the-art machine learning techniques, we
predict the experimental band gaps from computationally inexpensive density
functional theory calculations. We propose a modified Behler-Parrinello (BP)
architecture that greatly improves the model capacity while maintaining the
symmetry properties of the BP architecture. Using this scheme, we obtain band
gaps at a level of accuracy comparable to those obtained with state of the art
and computationally intensive hybrid functionals, thus significantly reducing
the computational cost of the task.
