  The solution of complex many-body lattice models can often be found by
defining an energy functional of the relevant density of the problem. For
instance, in the case of the Hubbard model the spin-resolved site occupation is
enough to describe the system total energy. Similarly to standard density
functional theory, however, the exact functional is unknown and suitable
approximations need to be formulated. By using a deep-learning neural network
trained on exact-diagonalization results we demonstrate that one can construct
an exact functional for the Hubbard model. In particular, we show that the
neural network returns a ground-state energy numerically indistinguishable from
that obtained by exact diagonalization and, most importantly, that the
functional satisfies the two Hohenberg-Kohn theorems: for a given ground-state
density it yields the external potential and it is fully variational in the
site occupation.
