  We train a neural network as the universal exchange-correlation functional of
density-functional theory that simultaneously reproduces both the exact
exchange-correlation energy and potential. This functional is extremely
non-local, but retains the computational scaling of traditional local or
semi-local approximations. It therefore holds the promise of solving some of
the delocalization problems that plague density-functional theory, while
maintaining the computational efficiency that characterizes the Kohn-Sham
equations. Furthermore, by using automatic differentiation, a capability
present in modern machine-learning frameworks, we impose the exact mathematical
relation between the exchange-correlation energy and the potential, leading to
a fully consistent method. We demonstrate the feasibility of our approach by
looking at one-dimensional systems with two strongly-correlated electrons,
where density-functional methods are known to fail, and investigate the
behavior and performance of our functional by varying the degree of
non-locality.
