We review recent advances in machine learning (ML) force-field methods for Landau-Lifshitz-Gilbert (LLG) simulations of itinerant electron magnets, focusing on scalability and transferability. Built on the principle of locality, a deep neural network model is developed to efficiently and accurately predict the electron-mediated forces governing spin dynamics. Symmetry-aware descriptors constructed through a group-theoretical approach ensure rigorous incorporation of both lattice and spin-rotation symmetries. The framework is demonstrated using the prototypical s-d exchange model widely employed in spintronics. ML-enabled large-scale simulations reveal novel nonequilibrium phenomena, including anomalous coarsening of tetrahedral spin order on the triangular lattice and the freezing of phase separation dynamics in lightly hole-doped, strong-coupling square-lattice systems. These results establish ML force-field frameworks as scalable, accurate, and versatile tools for modeling nonequilibrium spin dynamics in itinerant magnets.
Figures & tables
Figure 1: Schematic diagram of ML force-field model for itinerant electron magnets. A descriptor transforms the neighborhood spin configuration Ci to effective coordinates {prΓ,ηrΓ} which are then fed into a neural network (NN). The output node of the NN corresponds to the local energy ϵi=ε(Ci) associated with spin Si . The corresponding total potential energy E is obtained from the summation of these local energies. Automatic differentiation is employed to compute the derivatives ∂E/∂Si from which the local exchange fields Hi are obtained.
Figure 2: (a) Tetrahedral spin order on a triangular lattice [Eq. ( 9 )]. (b) Two inequivalent spins in a quadrupled unit cell with opposite chirality. (c) Comparison of ML-predicted torque TML with KPM results TKPM , where Ti=Si×Hi . (d) Histogram of prediction error δ=TKPM−TML .
Figure 3: (a) Snapshots of local scalar chirality χijk=Si⋅Sj×Sk , with (ijk) denoting sites of an elementary triangle, at different times after a thermal quench of the triangular s–d model near n=1/4 . (b) Characteristic length L versus time, extracted from the chirality structure factor (linear scales on both axes). Yellow triangles and green circles show data from LLG quenches using KPM and ML-calculated fields, respectively. The ML model was trained only on KPM–LLG data up to ttraining=800 , marked by the yellow shaded region.
Figure 4: (a) ML-predicted exchange fields versus exact results from the test dataset. (b) The distribution of the force difference δ=HML−Hexact , well fitted by a normal distribution (red line) with variance σ2=0.035 . (c) Spin-spin correlation ⟨Si⋅Sj⟩ as a function of distance rij=∣rj−ri∣ along the x direction at filling f=0.485 . Red dots show LLG results using NN models without Langevin noise, while blue lines correspond to ED-LLG simulations at T=0.022 .
Figure 5: (a) Density maps of local bond variables b(ri) at four different times during the ML–LLG simulation on a 100×100 lattice with 1.5% hole doping. (b) Average linear size L=⟨s⟩1/2 of FM clusters versus time after a thermal quench. The dash-dotted line denotes the t1/3 power-law growth, while the dashed line shows a sublogarithmic dependence L(t)∼(logt)β with β=0.11 . The inset plots the time evolution of the average cluster size ⟨s⟩ .
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∘ 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.
Gia-Wei Chern, Yunhao Fan, Sheng Zhang +1
Department of Physics, University of Virginia, Charlottesville, VA 22904, USA
Metallic magnets exhibit complex spin dynamics governed by electronically generated interactions. Predictive simulations of such dynamics typically require repeated solutions of an underlying electronic problem throughout the time evolution, creating a major computational bottleneck. Here we introduce a graph neural network (GNN) magnetic force-field framework that learns the effective magnetic energy functional governing itinerant spin dynamics directly from electronic calculations. Conceptually analogous to machine-learned interatomic potentials, the proposed framework enables efficient evaluation of spin torques while capturing the nonlinear and spatially extended interactions generated by itinerant electrons. We benchmark the method on representative metallic magnetic systems exhibiting collinear, noncollinear, and noncoplanar magnetic order. The learned force fields accurately reproduce electronically generated spin torques and yield nonequilibrium spin dynamics in excellent agreement with direct electronic simulations. Our results establish graph neural networks as a powerful framework for machine-learned magnetic force fields, providing a pathway toward predictive large-scale simulations of nonequilibrium magnetism across multiple length and time scales.
Ali Rayat, Yunhao Fan, Gia-Wei Chern
Department of Physics, University of Virginia, Charlottesville, VA 22904, USA
We present a magnetic extension of the Hierarchically Interacting Particle Neural Network (HIP-NN) that enables large-scale simulations of electron-mediated spin dynamics in disordered itinerant magnets. The resulting magnetic HIP-NN (mHIP-NN) incorporates rotationally invariant spin correlations directly into hierarchical message-passing layers, enabling the network to learn emergent magnetic energy landscapes and effective local fields from coupled geometric-spin environments while preserving spin-rotation symmetry. As a benchmark application, we consider structurally disordered itinerant s-d exchange models in which the effective magnetic forces arise dynamically from the instantaneous electronic structure and are computationally prohibitive to evaluate using conventional exact-diagonalization-based approaches. We show that mHIP-NN accurately reproduces the local torques governing Landau-Lifshitz-Gilbert dynamics and faithfully captures the nonequilibrium evolution of spatial spin correlations following thermal quenches. Our results establish symmetry-aware hierarchical message-passing networks as an efficient and scalable framework for large-scale simulations of frustrated itinerant spin systems and nonequilibrium magnetic dynamics. More broadly, because the learned energy functional remains fully differentiable with respect to both atomic coordinates and spin variables, the framework also provides a natural foundation for spin-dependent interatomic potentials and coupled atom-spin dynamics.
Supriyo Ghosh, Yunhao Fan, Sheng Zhang +2
Department of Physics, University of Virginia, Charlottesville, Virginia 22904, USA · Department of Chemistry, University of Chicago, Chicago, IL 60637, USA · Theoretical Division and CNLS, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA