  We developed a deep generative model-based variational free energy approach
to the equations of state of dense hydrogen. We employ a normalizing flow
network to model the proton Boltzmann distribution and a fermionic neural
network to model the electron wave function at given proton positions. By
jointly optimizing the two neural networks we reached a comparable variational
free energy to the previous coupled electron-ion Monte Carlo calculation. The
predicted equation of state of dense hydrogen under planetary conditions is
denser than the findings of ab initio molecular dynamics calculation and
empirical chemical model. Moreover, direct access to the entropy and free
energy of dense hydrogen opens new opportunities in planetary modeling and
high-pressure physics research.
