  Monolayer and multilayer graphene are promising materials for applications
such as electronic devices, sensors, energy generation and storage, and
medicine. In order to perform large-scale atomistic simulations of the
mechanical and thermal behavior of graphene-based devices, accurate interatomic
potentials are required. Here, we present a new interatomic potential for
multilayer graphene structures referred to as "hNN--Gr$_x$." This hybrid
potential employs a neural network to describe short-range interactions and a
theoretically-motivated analytical term to model long-range dispersion. The
potential is trained against a large dataset of monolayer graphene, bilayer
graphene, and graphite configurations obtained from ab initio total-energy
calculations based on density functional theory (DFT). The potential provides
accurate energy and forces for both intralayer and interlayer interactions,
correctly reproducing DFT results for structural, energetic, and elastic
properties such as the equilibrium layer spacing, interlayer binding energy,
elastic moduli, and phonon dispersions to which it was not fit. The potential
is used to study the effect of vacancies on thermal conductivity in monolayer
graphene and interlayer friction in bilayer graphene. The potential is
available through the OpenKIM interatomic potential repository at
\url{https://openkim.org}.
