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.
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Figure 1: Schematic diagram of ML force-field model for itinerant electron magnets. A descriptor transforms the neighborhood spin configuration Ci to effective coordinates {Gℓ} which are then fed into a neural network (NN). The output node of the NN correspond to the local energy ϵi=ε(Ci) associated with spin Si . {xm(1)} and {xn(2)} denote feature variables associated with neurons of the first and second hidden layers, respectively. The corresponding total potential energies E is obtained from 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) The four bond variables bA,bB,bC,bD , defined as the inner products between the central spin Si and its four equidistant nearest neighbors, form the basis of a four-dimensional reducible representation of the D4 point group. (b) The four scalar chiralities χA,χB,χC,χD likewise span a four-dimensional reducible representation of D4 . (c) The eight bond variables bA,bB,…,bH constitute the basis of an eight-dimensional reducible representation.
Figure 3: Top panels: (a) The non-collinear 120 ∘ magnetic order with the tripled unit cell of the half-filled triangular-lattice s–d model. (b) Benchmark of ML-predicted torques, TML=S×HML , evaluated for the 120 ∘ state. (c) Ensemble-averaged spin structure factor S(q) obtained from ML-LLG simulations. Bottom panels: Results for the non-coplanar tetrahedral order at filling fraction f≈1/4 : (d) real-space spin configuration with a quadrupled magnetic unit cell; (e) ML torque-prediction benchmark; and (f) the corresponding ensemble-averaged spin structure factor.
Figure 4: (a) ML-predicted exchange fields compared against exact values from the test dataset. (b) Distribution of the force difference δ=HML−Hexact , which is well described by a normal distribution (red curve) with variance σ2=0.035 .
Figure 5: (Top) Density plots of the local nearest-neighbor spin correlation bi≡(Si⋅Si+x+Si⋅Si−x+Si⋅Si+y+Si⋅Si−y)/4 at four representative times during the ML-LLG simulation on a 200×200 lattice. (Bottom) Corresponding ML-predicted distributions of the local electron number ni=∑α⟨c^iα†c^iα⟩ . The neural-network model is trained on datasets obtained from exact solutions for an L=30 lattice with parameters tnn=1 , J=7 , and electron filling fraction n=0.485 .
Figure 6: Average linear size L=⟨s⟩1/2 of ferromagnetic (FM) clusters as a function of time following a thermal quench. The dash–dot line marks the power-law growth L(t)∼t1/3 , while the dashed line shows a sublogarithmic form L(t)∼(logt)β with β=0.11 . The inset displays the corresponding time evolution of the average cluster size ⟨s⟩ .
Figure 7: A tangential vector field on a sphere can be decomposed into (a) the curl-free component ∇sE and (b) the divergence-free component ∇s×G . (c) shows the gradient field hieq=−∂E/∂Si , which can be viewed as a quasi-equilibrium exchange field, and the curl-field hineq=−Si×∂G/∂Si , which corresponds to the nonequilibrium force, and their respective torques Tieq=hieq×Si and Tineq=hineq×Si .
Figure 8: (a) Schematic diagram of neural-network (NN) force-field model for out-of-equilibrium itinerant spin system. A descriptor transforms the neighborhood spin configuration Ci to effective coordinates {Gℓ} which are then fed into a NN. The two output nodes of the NN correspond to the local energy ϵi=ε(Ci) and γi=χ(Ci) associated with site- i . The corresponding total potential energies E and G are obtained from summation of these local energies. Automatic differentiation is employed to compute the derivatives ∂E/∂Si and ∂G/∂Si , from which the local exchange fields Hi are obtained according to the generalized force expression Eq.( 21 ).
Figure 9: (Color online) Schematic diagram of the voltage-driven magnetic transition in the DE model sandwiched by two electrodes. In the initial state, spins in the DE model are arranged in a staggered Néel order. A voltage drop V is applied to the two electrodes.
Figure 10: (a) The ML predicted forces versus the exact solution from the NEGF calculation for the s-d model with exchange coupling J=3.8tnn . (b) Normalized distribution of the prediction error of the perpendicular components of the forces.
Figure 11: Domain wall propagation in a s-d system driven by an external voltage eV=3.2tnn . Comparison between (a) NEGF-LLG simulations and (b) ML-LLG simulations. The lattice size is 30×24 . The color bar indicates the local nearest-neighbor spin correlation bij=Si⋅Sj . Blue (red) area corresponds to AFM (FM) domains. The NEGF-LLG simulation was carried out at a low temperature of kBT=0.01tnn , while the ML simulation was performed without Langevin noise.
Figure 12: (a) Average position of FM-AFM domain, obtained from NEGF-LLG and ML-LLG simulations, as a function of time during the voltage-driven insulator to metal transition of the s-d model. (b) histogram of the ratio ∣Tineq∣/∣Tieq∣ predicted by the NN model for the simulation of domain-wall propagation. Here Tieq=Si×∂E/∂Si is the quasi-equilibrium torque, and Tineq=Si×(Si×∂G/∂Si) is the non-equilibrium torque in the generalized LLG Eq. ( VI.1 ).
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.
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