  Adsorption of carbon monoxide (CO) on transition-metal surfaces is a
prototypical process in surface sciences and catalysis. Despite its simplicity,
it has posed great challenges to theoretical modeling. Pretty much all existing
density functionals fail to accurately describe surface energies, CO adsorption
site preference, as well as adsorption energies simultaneously. Although the
random phase approximation (RPA) cures these density functional theory
failures, its large computational cost makes it prohibitive to study the CO
adsorption for any but the simplest ordered cases. Here, we address these
challenges by developing a machine-learned force field (MLFF) with near RPA
accuracy for the prediction of coverage-dependent adsorption of CO on the
Rh(111) surface through an efficient on-the-fly active learning procedure and a
$\Delta$-machine learning approach. We show that the RPA-derived MLFF is
capable to accurately predict the Rh(111) surface energy, CO adsorption site
preference as well as adsorption energies at different coverages that are all
in good agreement with experiments. Moreover, the coverage-dependent
ground-state adsorption patterns and adsorption saturation coverage are
identified.
