  We describe the development of machine-learned potentials of atmospheric
gases with flexible monomers for molecular simulations. A recently suggested
permutationally invariant polynomial neural network (PIP-NN) approach is
utilized to represent the full-dimensional two-body component of the dimer
energy. To ensure the asymptotic zero-interaction limit, a tailored subset of
the full invariant polynomial basis set is utilized and their variables are
modified to achieve a better fit of the correct asymptotic behavior at a long
range. The new technique is used to build full-dimensional potentials for the
two-body N$_2-$Ar and N$_2-$CH$_4$ interactions by fitting databases of ab
initio energies calculated at the coupled-cluster level of theory. The second
virial coefficients with full account of molecular flexibility effects are then
calculated within the classical framework using the PIP-NN potential surfaces.
To showcase the advantages of the PIP-NN method, we compare its accuracy and
computational efficiency to several kernel-based and neural-network-based
approaches using the MD17 database of energies and forces for ethanol. For
large training set sizes, the PIP-NN models attain the best accuracy among
examined models, and the computation time is shown to be comparable to that of
the PIP regression model and several orders of magnitude faster than the
quickest alternatives.
