  Self-learning hybrid Monte Carlo (SLHMC) is a first-principles simulation
that allows for exact ensemble generation on potential energy surfaces based on
density functional theory. The statistical sampling can be accelerated with the
assistance of smart trial moves by machine learning potentials. In the first
report (Nagai, {\it et al}. Phys. Rev. B 102, 041124(R) (2020)), the SLHMC
approach was introduced for the simplest case of canonical sampling. We herein
extend this idea to isothermal-isobaric ensembles to enable general
applications for soft materials and liquids with large volume fluctuation. As a
demonstration, the isothermal-isobaric SLHMC method was used to study the
vibrational structure of liquid silica at temperatures close to the melting
point, whereby the slow diffusive motion is beyond the time scale of
first-principles molecular dynamics. It was found that the static structure
factor thus computed from first-principles agrees quite well with the
high-energy X-ray data.
