  Boltzmann generators (BGs) are now recognized as forefront generative models
for sampling equilibrium states of many-body systems in the canonical ensemble,
as well as for calculating the corresponding Helmholtz free energy.
Furthermore, BGs can potentially provide a notable improvement in efficiency
compared to conventional techniques such as molecular dynamics (MD) and Monte
Carlo (MC) methods. By sampling from a clustered latent space, BGs can
circumvent free-energy barriers and overcome the rare-event problem. However,
one major limitation of BGs is their inability to sample across phase
transitions between ordered phases. This is due to the fact that new phases may
not be commensurate with the box dimensions, which remain fixed in the
canonical ensemble. In this work, we present a novel BG model for the
isothermal-isobaric (NPT) ensemble, which can successfully overcome this
limitation. This unsupervised machine-learning model can sample equilibrium
states at various pressures, as well as pressure-driven phase transitions. We
demonstrate that the samples generated by this model are in good agreement with
those obtained through MD simulations of two model systems. Additionally, we
derive an estimate of the Gibbs free energy using samples generated by the NPT
BG.
