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This study introduces time-reversal E(3)-equivariant neural network andSpinGNN++ framework for constructing a comprehensive interatomic potential formagnetic systems, encompassing spin-orbit coupling and noncollinear magneticmoments. SpinGNN++ integrates multitask spin equivariant neural network withexplicit spin-lattice terms, including Heisenberg, Dzyaloshinskii-Moriya,Kitaev, single-ion anisotropy, and biquadratic interactions, and employstime-reversal equivariant neural network to learn high-order spin-latticeinteractions using time-reversal E(3)-equivariant convolutions. To validateSpinGNN++, a complex magnetic model dataset is introduced as a benchmark andemployed to demonstrate its capabilities. SpinGNN++ provides accuratedescriptions of the complex spin-lattice coupling in monolayer CrI$_3$ andCrTe$_2$, achieving sub-meV errors. Importantly, it facilitates large-scaleparallel spin-lattice dynamics, thereby enabling the exploration of associatedproperties, including the magnetic ground state and phase transition.Remarkably, SpinGNN++ identifies a new ferrimagnetic state as the groundmagnetic state for monolayer CrTe2, thereby enriching its phase diagram andproviding deeper insights into the distinct magnetic signals observed invarious experiments.

General time-reversal equivariant neural network potential for magnetic materials