Local-energy-driven machine-learning exchange field and longitudinal spin fluctuation

Understanding and efficiently modeling noncollinear spin dynamics requires accurate representations of both magnetic energies and their derivatives with respect to spin orientations. Building on the Smooth Overlap of Spin Orientations (SOSO) descriptor and Gaussian Approximation Potential framework introduced in our previous work [Gao et al., Phys. Rev. B 113, 144413 (2026)], we investigate two extensions of the learning strategy: site-resolved magnetic-energy supervision and environment-dependent prediction of magnetic-moment magnitudes. Using a short-range Heisenberg model for bcc Fe, we show that training on local atomic magnetic energies greatly improves the data efficiency and the accuracy of magnetic-energy and transverse exchange-field predictions compared with training only on the total energy. The model is further used to learn longitudinal variations in the magnetic moments from their local spin environments. Constrained noncollinear density functional theory calculations for bcc Fe are employed separately to test the capability of the SOSO representation to describe first-principles noncollinear magnetic configurations and local magnetic moments. As site-resolved magnetic energies are not presently available from our constrained-density functional theory workflow, the local-energy training strategy demonstrated here should be regarded as a proof of concept at the Heisenberg model level. These results clarify the potential benefits of local magnetic supervision and provide a basis for future first-principles extensions of machine-learning spin-dynamics models.