  The advances of machine-learned force fields have opened up molecular
dynamics (MD) simulations for compounds for which ab-initio MD is too
resource-intensive and phenomena for which classical force fields are
insufficient. Here we describe a neural-network force field parametrized to
reproduce the r2SCAN potential energy landscape of HfO$_2$. Based on an
automatic differentiable implementation of the isothermal-isobaric (NPT)
ensemble with flexible cell fluctuations, we study the phase space of HfO$_2$.
We find excellent predictive capabilities regarding the lattice constants and
experimental X-ray diffraction data. The phase transition away from monoclinic
is clearly visible at a temperature around 2000 K, in agreement with available
experimental data and previous calculations. Another abrupt change in lattice
constants occurs around 3000 K. While the resulting lattice constants are
closer to cubic, they exhibit a small tetragonal distortion, and there is no
associated change in volume. We show that this high-temperature structure is in
agreement with the available high-temperature diffraction data.
