  Neural networks (NNs) are employed to predict equations of state from a given
isotropic pair potential using the virial expansion of the pressure. The NNs
are trained with data from molecular dynamics simulations of monoatomic gases
and liquids, sampled in the $NVT$ ensemble at various densities. We find that
the NNs provide much more accurate results compared to the analytic low-density
limit estimate of the second virial coefficient. Further, we design and train
NNs for computing (effective) pair potentials from radial pair distribution
functions, $g(r)$, a task which is often performed for inverse design and
coarse-graining. Providing the NNs with additional information on the forces
greatly improves the accuracy of the predictions, since more correlations are
taken into account; the predicted potentials become smoother, are significantly
closer to the target potentials, and are more transferable as a result.
