Machine-learning modeling of magnetization dynamics in quasi-equilibrium and driven metallic spin systems

We review recent advances in machine-learning (ML) force-field methods for large-scale Landau-Lifshitz-Gilbert (LLG) simulations of metallic spin systems. We generalize the Behler-Parrinello (BP) ML architecture -- originally developed for quantum molecular dynamics -- to construct scalable and transferable ML models capable of capturing the intricate dependence of electron-mediated exchange fields on the local magnetic environment characteristic of itinerant magnets. A central ingredient of this framework is the implementation of symmetry-aware magnetic descriptors based on group-theoretical bispectrum formalisms. Leveraging these ML force fields, LLG simulations faithfully reproduce hallmark non-collinear magnetic orders -- such as the $120^\circ$ and tetrahedral states -- on the triangular lattice, and successfully capture the complex spin textures emerging in the mixed-phase states of a square-lattice double-exchange model under thermal quench. We further discuss a generalized potential theory that extends the BP formalism to incorporate both conservative and nonconservative electronic torques, thereby enabling ML models to learn nonequilibrium exchange fields from computationally demanding microscopic approaches such as nonequilibrium Green's-function techniques. This extension yields quantitatively accurate predictions of voltage-driven domain-wall motion and establishes a foundation for quantum-accurate, multiscale modeling of nonequilibrium spin dynamics and spintronic functionalities.