  The Green-Kubo (GK) method is a rigorous framework for heat transport
simulations in materials. However, it requires an accurate description of the
potential-energy surface and carefully converged statistics. Machine-learning
potentials can achieve the accuracy of first-principles simulations while
allowing to reach well beyond their simulation time and length scales at a
fraction of the cost. In this paper, we explain how to apply the GK approach to
the recent class of message-passing machine-learning potentials, which
iteratively consider semi-local interactions beyond the initial interaction
cutoff. We derive an adapted heat flux formulation that can be implemented
using automatic differentiation without compromising computational efficiency.
The approach is demonstrated and validated by calculating the thermal
conductivity of zirconium dioxide across temperatures.
