  The discovery of ferroelectricity in HfO$_2$-based thin films opens up new
opportunities for using this silicon-compatible ferroelectric to realize
low-power logic circuits and high-density non-volatile memories. The functional
performances of ferroelectrics are intimately related to their dynamic
responses to external stimuli such as electric fields at finite temperatures.
Molecular dynamics is an ideal technique for investigating dynamical processes
on large length and time scales, though its applications to new materials is
often hindered by the limited availability and accuracy of classical force
fields. Here we present a deep neural network-based interatomic force field of
HfO$_2$ learned from {\em ab initio} data using a concurrent learning
procedure. The model potential is able to predict structural properties such as
elastic constants, equation of states, phonon dispersion relationships, and
phase transition barriers of various hafnia polymorphs with accuracy comparable
with density functional theory calculations. The validity of this model
potential is further confirmed by the reproduction of experimental sequences of
temperature-driven ferroelectric-paraelectric phase transitions of HfO$_2$ with
isobaric-isothermal ensemble molecular dynamics simulations. We suggest a
general approach to extend the model potential of HfO$_2$ to related material
systems including dopants and defects.
