Ab initio molecular dynamics (AIMD) refers to the solution of Newton's equations of motion for ions with forces ${\mathbf{f}}_i=-\partial E/\partial {\mathbf{R}}_i$ calculated from self-consistent electronic structure calculations. So-called machine-learning force field (ML-FF) schemes parametrize the potential energy surface very efficiently and make it possible to extend the time scale of AIMD simulations by orders of magnitude. The Landau-Lifshitz equation describes the dynamics of atomic magnetic moments ${\mathbf{m}}_i$ in effective fields ${\mathbf{h}}_i=-\partial E/\partial {\mathbf{m}}_i$ which in addition to containing external magnetic fields, describe contributions from interatomic exchange interactions, long-range dipolar interactions, anisotropy fields, etc. In this publication, we add the magnetic degrees of freedom to the widely used Gaussian approximation potential of machine learning and present a model that describes the potential energy surface of a crystal based on atomic coordinates and noncollinear magnetic moments. Incorporating the translational, rotational, inversion, and permutational symmetries of magnetic interactions, the ML model can describe various magnetic interactions expanded into two-body, three-body terms, etc., in the spirit of the atomic cluster expansion. Assuming an adiabatic approximation for the spin directions, the ML model depends solely on the positions and orientations of atomic spins and is computationally efficient enough to make coupled ab initio molecular and spin dynamics possible. To illustrate the ML model, we implement a two-body form for the interatomic exchange interaction. Comparing the total energies and local exchange fields predicted by the model for noncollinear spin arrangements with the results of constrained noncollinear density functional calculations for bcc Fe yields very good results, with agreement on the level of 1 meV/spin for the total energy.

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• Exchange interaction
• Spin dynamics
• Density functional calculations
• Machine learning