Machine-learned force fields (MLFFs) are increasingly used to accelerate first-principles simulations of many materials properties. However, MLFFs are generally trained from density functional theory (DFT) data and thus suffer from the same limitations as DFT. To achieve more predictive accuracy, MLFFs based on higher levels of theory are required, but the training becomes exceptionally arduous. Here, we present an approach to generate MLFFs with beyond DFT accuracy which combines an efficient on-the-fly active learning method and $\mathrm{\Delta }$-machine learning. Using this approach, we 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. We show that owing to the relatively smooth nature of these differences, the expensive RPA calculations can be performed only on a small number of representative structures of small unit cells selected by rank compression of the kernel matrix. This dramatically reduces the computational cost and allows one 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. These results open the way to many-body calculations of finite-temperature properties of materials.

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• Phase transitions
• Transition temperature
• Insulators
• First-principles calculations
• Machine learning
• Molecular dynamics
• Random phase approximation