  Phonon-based approaches and molecular dynamics are widely established methods
for gaining access to a temperature-dependent description of material
properties. However, when a compound's phase space is vast,
density-functional-theory-backed studies quickly reach prohibitive levels of
computational expense. Here, we explore the complex phase structure of HfO2
using effective harmonic potentials based on a neural-network force field
(NNFF) as a surrogate model. We detail the data acquisition and training
strategy that enable the NNFF to provide almost ab-initio accuracy at a
significantly reduced cost and present a recipe for automation. We demonstrate
how the NNFF can generalize beyond its training data and that it is
transferable between several phases of hafnia. We find that the thermal
expansion of the low-symmetry phases agrees well with experimental results and
we determine the P-43m phase to be the favorable (stoichiometric) cubic phase
over the established Fm-3m. In contrast, the experimental lattice constants of
the cubic phases are substantially larger than what is calculated for the
corresponding stoichiometric phases. Furthermore, we show that the
stoichiometric cubic phases are unlikely to be thermodynamically stable
compared to the tetragonal and monoclinic phases, and hypothesize that they
only exist in defect-stabilized forms.
