Machine-learning potentials are usually trained on the ground-state, Born-Oppenheimer energy surface, which depends exclusively on the atomic positions and not on the simulation temperature. This disregards the effect of thermally excited electrons, that is important in metals, and essential to the description of warm dense matter. An accurate physical description of these effects requires that the nuclei move on a temperature-dependent electronic free energy. We propose a method to obtain machine-learning predictions of this free energy at an arbitrary electron temperature using exclusively training data from ground-state calculations, avoiding the need to train temperature-dependent potentials, and benchmark it on metallic liquid hydrogen at the conditions of the core of gas giants and brown dwarfs. This Letter demonstrates the advantages of hybrid schemes that use physical consideration to combine machine-learning predictions, providing a blueprint for the development of similar approaches that extend the reach of atomistic modeling by removing the barrier between physics and data-driven methodologies.

This record contains the raw outputs of the DFT calculations done on the training set. These files are denoted by the "training set folder #*" description. The record also contains a minimal working example showing the ML workflow for training the data and how to run the MD simulations. We include the training data in the format of XYZ files for the structures and NumPy arrays for the DFT energies, forces and DOS. We also provide a Chemiscope visualisation file of the training set.