  We combine machine learning (ML) with Monte Carlo (MC) simulations to study
the crystal nucleation process. Using ML, we evaluate the canonical partition
function of the system over the range of densities and temperatures spanned
during crystallization. We achieve this on the example of the Lennard-Jones
system by training an artificial neural network using, as a reference dataset,
equations of state for the Helmholtz free energy for the liquid and solid
phases. The accuracy of the ML predictions is tested over a wide range of
thermodynamic conditions, and results are shown to provide an accurate estimate
for the canonical partition function, when compared to the results from
flat-histogram simulations. Then, the ML predictions are used to calculate the
entropy of the system during MC simulations in the isothermal-isobaric
ensemble. This approach is shown to yield results in very good agreement with
the experimental data for both the liquid and solid phases of Argon. Finally,
taking entropy as a reaction coordinate and using the umbrella sampling
technique, we are able to determine the Gibbs free energy profile for the
crystal nucleation process. In particular, we obtain a free energy barrier in
very good agreement with the results from previous simulation studies. The
approach developed here can be readily extended to molecular systems and
complex fluids, and is especially promising for the study of entropy-driven
processes.
