  Machine learning is a powerful tool to design accurate, highly non-local,
exchange-correlation functionals for density functional theory. So far, most of
those machine learned functionals are trained for systems with an integer
number of particles. As such, they are unable to reproduce some crucial and
fundamental aspects, such as the explicit dependency of the functionals on the
particle number or the infamous derivative discontinuity at integer particle
numbers. Here we propose a solution to these problems by training a neural
network as the universal functional of density-functional theory that (i)
depends explicitly on the number of particles with a piece-wise linearity
between the integer numbers and (ii) reproduces the derivative discontinuity of
the exchange-correlation energy. This is achieved by using an ensemble
formalism, a training set containing fractional densities, and an explicitly
discontinuous formulation.
