cond-mat.mes-hallAug 22, 2026

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

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Abstract

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 0.004250.00425 with norm drift at the 10−710^{-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\mathbb{S}^{2} , 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 0.45 ns0.45\,\mathrm{ns} .
Evaluation regimeaJHa_{JH} rangeRMSEMaximum norm drift
In-domain[−0.08,−0.03][-0.08,-0.03]0.00425±0.001260.00425\pm 0.001261.788×10−71.788\times 10^{-7}
OOD[−0.10,−0.08001][-0.10,-0.08001]0.0259±0.006560.0259\pm 0.006561.192×10−71.192\times 10^{-7}
Table 1: Long-horizon closed-loop performance of the Flow model over five independent training runs. RMSE is reported as mean ±\pm 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=−0.10a_{JH}=-0.10 . Closed-loop propagation begins at t=99.99 nst=99.99\,\mathrm{ns} and preserves unit norm throughout. Relative to the reference flow, the predicted asymptotic direction is systematically shifted in the mxm_{x} and mym_{y} components.
Model variantIn-domain RMSEOOD RMSEIn-domain maximum norm drift
Full model: total data weight 2.0 + comp(0.1) + retraction0.0065990.0301871.788×10−71.788\times 10^{-7}
Remove composition0.0070050.0312241.788×10−71.788\times 10^{-7}
Duplicate-term weight 1.0→0.51.0\rightarrow 0.5 (total data weight 2.0→1.52.0\rightarrow 1.5 )0.0068170.0304821.788×10−71.788\times 10^{-7}
Remove duplicate term (total data weight 2.0→1.02.0\rightarrow 1.0 )0.0074060.0330671.788×10−71.788\times 10^{-7}
Retraction →\rightarrow exponential map0.0075390.0363091.192×10−71.192\times 10^{-7}
Retraction →\rightarrow raw output0.0073210.0369935.065×10−35.065\times 10^{-3}
Table 2: Controlled single-factor ablations. All variants share initialization and minibatch order; each result is from one paired run.
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