  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 work
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 modelling
by removing the barrier between physics and data-driven methodologies.
