# Physics-Constrained Neural Flow Maps for Long-Horizon Prediction of Spin Dynamics

arXiv: https://arxiv.org/abs/2608.22006 · HTML: https://mlsanity.com/paper/2608.22006/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 4 of its 31 references that are in the map.
- First posted on arXiv 22 Aug 2026.

## Builds on (most cited first, 4 of 4)

- [Data Driven Governing Equations Approximation Using Deep Neural Networks](https://mlsanity.com/paper/1811.05537/) (2018)
- [Flow Map Learning for Unknown Dynamical Systems: Overview, Implementation, and Benchmarks](https://mlsanity.com/paper/2307.11013/) (2023)
- [Deep-OSG: Deep Learning of Operators in Semigroup](https://mlsanity.com/paper/2302.03358/) (2023)
- [Learning Hamiltonian Flow Maps: Mean Flow Consistency for Large-Timestep Molecular Dynamics](https://mlsanity.com/paper/2601.22123/) (2026)

## Abstract (arXiv)

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.