  Hydrous phases play a fundamental role in the deep-water cycle on Earth.
Understanding their stability and thermoelastic properties is essential for
constraining their abundance using seismic tomography. However, determining
their elastic properties at extreme conditions is notoriously challenging. The
challenges stem from the complex behavior of hydrogen bonds under high
pressures and temperatures (P,Ts). In this study, we evaluate how advanced
molecular dynamics simulation techniques can address these challenges by
investigating the adiabatic elasticity and acoustic velocities of
$\delta$-AlOOH, a critical and prototypical high-pressure hydrous phase. We
compared the performances of three methods to assess their viability and
accuracy. The thermoelastic tensor was computed up to 140 GPa and temperatures
up to 2,700 K using molecular dynamics with a DeePMD machine-learning
interatomic potential based on the SCAN meta-GGA functional. The excellent
agreement with ambient condition single-crystal ultrasound measurements and the
correct description of velocity changes induced by H-bond
disorder-symmetrization transition observed at 10 GPa in Brillouin scattering
measurements underscores the accuracy and efficacy of our approach.
