Machine learning nonequilibrium electron forces for spin dynamics of itinerant magnets

We present a generalized potential theory for conservative as well as nonconservative forces for the Landau-Lifshitz magnetization dynamics. Importantly, this formulation makes possible an elegant generalization of the Behler-Parrinello machine learning (ML) approach, which is a cornerstone of ML-based quantum molecular dynamics methods, to the modeling of force fields in adiabatic spin dynamics of out-of-equilibrium itinerant magnetic systems. We demonstrate our approach by developing a deep-learning neural network that successfully learns the electron-mediated exchange forces in a driven s-d model computed from the nonequilibrium Green's function method. We show that dynamical simulations with forces predicted from the neural network accurately reproduce the voltage-driven domain-wall propagation. Our work also lays the foundation for ML modeling of spin transfer torques and opens a new avenue for ML-based multi-scale modeling of nonequilibrium dynamical phenomena in itinerant magnets and spintronics.