We propose a machine-learning interatomic potential for multi-component magnetic materials. In this potential we consider magnetic moments as degrees of freedom (features) along with atomic positions, atomic types, and lattice vectors. We create a training set with constrained DFT (cDFT) that allows us to calculate energies of configurations with non-equilibrium (excited) magnetic moments and, thus, it is possible to construct the training set in a wide configuration space with great variety of non-equilibrium atomic positions, magnetic moments, and lattice vectors. Such a training set makes possible to fit reliable potentials that will allow us to predict properties of configurations in the excited states (including the ones with non-equilibrium magnetic moments). We verify the trained potentials on the system of bcc Fe-Al with different concentrations of Al and Fe and different ways Al and Fe atoms occupy the supercell sites. Here, we show that the formation energies, the equilibrium lattice parameters, and the total magnetic moments of the unit cell for different Fe-Al structures calculated with machine-learning potentials are in good correspondence with the ones obtained with DFT. We also demonstrate that both the machine-learning interatomic potentials and DFT qualitatively reproduce the experimentally-observed anomalous volume-composition dependence in the Fe-Al system.

This work was supported by Russian Science Foundation (grant number 22-73-10206, https://rscf.ru/project/22-73-10206/). M.H. gratefully acknowledges the financial support under the scope of the COMET program within the K2 Center ``Integrated Computational Material, Process and Product Engineering (IC-MPPE)'' (Project No 886385).