  We present an approach to generate machine-learned force fields (MLFF) with
beyond density functional theory (DFT) accuracy. Our approach combines
on-the-fly active learning and $\Delta$-machine learning in order to generate
an MLFF for zirconia based on the random phase approximation (RPA).
Specifically, an MLFF trained on-the-fly during DFT based molecular dynamics
simulations is corrected by another MLFF that is trained on the differences
between RPA and DFT calculated energies, forces and stress tensors. Thanks to
the relatively smooth nature of the differences, the expensive RPA calculations
are performed only on a small number of representative structures of small unit
cells. These structures are determined by a singular value decomposition rank
compression of the kernel matrix with low spatial resolution. This dramatically
reduces the computational cost and allows us to generate an MLFF fully capable
of reproducing high-level quantum-mechanical calculations beyond DFT. We
carefully validate our approach and demonstrate its success in studying the
phase transitions of zirconia.
