Physics-Constrained Neural Flow Maps for Long-Horizon Prediction of Spin Dynamics
Authors: Haoen Feng, Shenglan Yuan, Shirong Lin
Organizations: School of Physical Sciences, Great Bay University, Dongguan 523000, China · School of Intelligent Systems Science and Engineering, Jinan University, Zhuhai 519000, China · Department of Mathematics, School of Sciences, Great Bay University, Dongguan 523000, China · Great Bay Institute for Advanced Study, Dongguan 523000, China
Conventional simulation of current-driven magnetization relies on fine-step integration of the spin-transfer-torque Landau--Lifshitz--Gilbert equation, creating a computational bottleneck in parameter sweeps and control searches. In this work, we propose a physics-constrained neural flow map that learns finite-time dynamics directly on the unit sphere. The model maps the current magnetization, spin-torque strength, and requested time span to a future state in a single forward pass. Tangent-space projection and spherical retraction preserve unit magnetization during recursive, composition-consistent rollout. We validate the framework on single-spin trajectories under in-domain torques and previously unseen but stronger drive. Beyond the training horizon, it achieves an in-domain root mean square error of 0.00425 with norm drift at the 10−7 level. The flow outperforms an adapted Long Short-Term Memory (LSTM) in in-domain accuracy and geometric stability, although the LSTM retains slightly lower out-of-distribution state error. The resulting geometry-preserving propagator reduces reliance on fine-step integration and enables physically admissible long-horizon prediction.
Figures & tables
Figure 1: Physics-constrained neural flow map. The current magnetization, normalized spin-transfer-torque strength, and requested prediction span are encoded as eight input features. A residual multi-layer perceptron produces a three-dimensional increment; tangent-space projection and spherical retraction constrain the predicted state to S2 , after which the prediction is fed back for closed-loop propagation. Training combines data fitting, duplicate supervision equivalent to the data term, and single-split composition consistency. Each closed-loop call spans ten fine steps, corresponding to 0.45ns .
Evaluation regime
aJH range
RMSE
Maximum norm drift
In-domain
[−0.08,−0.03]
0.00425±0.00126
1.788×10−7
OOD
[−0.10,−0.08001]
0.0259±0.00656
1.192×10−7
Table 1: Long-horizon closed-loop performance of the Flow model over five independent training runs. RMSE is reported as mean ± sample standard deviation. Maximum norm drift is the maximum over the complete evaluation set and time window.
Figure 2: Closed-loop comparison between Flow and the adapted LSTM over five independent training runs. The left panel shows componentwise RMSE for in-domain testing and stronger-drive OOD evaluation; the right panel shows maximum norm drift on a logarithmic scale. Bars and error bars denote the mean and sample standard deviation over five runs. Both methods use the same parameter split and future window, but LSTM warms up its hidden state with the complete observed history, whereas Flow reads only the boundary state.
Figure 4: Joint temporal and parameter extrapolation under the unseen, stronger drive aJH=−0.10 . Closed-loop propagation begins at t=99.99ns and preserves unit norm throughout. Relative to the reference flow, the predicted asymptotic direction is systematically shifted in the mx and my components.
Model variant
In-domain RMSE
OOD RMSE
In-domain maximum norm drift
Full model: total data weight 2.0 + comp(0.1) + retraction
0.006599
0.030187
1.788×10−7
Remove composition
0.007005
0.031224
1.788×10−7
Duplicate-term weight 1.0→0.5 (total data weight 2.0→1.5 )
0.006817
0.030482
1.788×10−7
Remove duplicate term (total data weight 2.0→1.0 )
0.007406
0.033067
1.788×10−7
Retraction → exponential map
0.007539
0.036309
1.192×10−7
Retraction → raw output
0.007321
0.036993
5.065×10−3
Table 2: Controlled single-factor ablations. All variants share initialization and minibatch order; each result is from one paired run.
Flow map learning trains neural models to predict finite-time state or distribution transitions directly, enabling fast simulation, control, and generation.
Long-horizon micromagnetic simulation remains expensive because conventional and learned solvers typically propagate Landau--Lifshitz--Gilbert (LLG) dynamics step by step. Existing learned approaches generally retain stepwise integration or model deterministic evolution, leaving full-field, direct-horizon stochastic prediction largely unexplored. We propose FLARE, a flow-matching framework that recasts stochastic finite-time magnetization prediction as conditional transport over anchor-relative local axis-angle rotations. This rotation-space formulation respects the intrinsic geometry of magnetization dynamics and preserves pointwise unit norm by construction. By explicitly conditioning on the physical prediction horizon, FLARE directly generates full-field stochastic endpoints across multiple target times without stepwise integration. Against the strongest single-checkpoint external baseline on each metric, FLARE achieves 29.9% lower angular energy distance (15.30∘), and a 37.3% lower fair energy score (0.393). On a representative composed 5-ns two-segment protocol, FLARE achieves a 3,062× best-batch speedup over the widely used GPU micromagnetic solver MuMax3 on a single GPU.
Pengyu Li, Renjie Tong, Xuanlue Jiang +2
State Key Laboratory of Quantum Functional Materials, Department of Physics, and Guangdong Basic Research Center of Excellence for Quantum Science, Southern University of Science and Technology, Shenzhen 518055, China
Simulating the long-time evolution of Hamiltonian systems is limited by the small timesteps required for stable numerical integration. To overcome this constraint, we introduce a framework to learn Hamiltonian Flow Maps by predicting the mean phase-space evolution over a chosen time span, enabling stable large-timestep updates far beyond the stability limits of classical integrators. To this end, we impose a Mean Flow consistency condition for time-averaged Hamiltonian dynamics. Unlike prior approaches, this allows training on independent phase-space samples without access to future states, avoiding expensive trajectory generation. Validated across diverse Hamiltonian systems, our method in particular improves upon molecular dynamics simulations using machine-learned force fields (MLFF). Our models maintain comparable training and inference cost, but support significantly larger integration timesteps while trained directly on widely-available trajectory-free MLFF datasets.
Winfried Ripken, Michael Plainer, Gregor Lied +5
Technical University Berlin · BIFOLD · Free University of Berlin +6
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