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 $120^\circ$ 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.

翻译：我们回顾了近期利用机器学习（ML）力场方法对金属自旋系统进行大规模Landau-Lifshitz-Gilbert（LLG）模拟的研究进展。我们将最初为量子分子动力学开发的Behler-Parrinello（BP）机器学习架构推广至可扩展且具有可迁移性的ML模型，使其能够捕捉局域磁环境中电子介导交换场对巡游磁体典型特征的复杂依赖关系。该框架的核心要素是基于群论双谱形式实现对称性感知的磁性描述符。借助这些机器学习力场，LLG模拟忠实再现了三角晶格上标志性的非共线磁有序（例如120°态和四面体态），并成功捕捉了方晶格双交换模型在热淬火过程中混合相态下涌现的复杂自旋织构。我们进一步讨论了一种广义势理论，该理论将BP形式扩展至同时包含保守与非保守电子扭矩，从而使ML模型能够从计算密集型的微观方法（如非平衡格林函数技术）中学习非平衡交换场。这一扩展为电压驱动畴壁运动提供了定量精确的预测，并为非平衡自旋动力学及自旋电子学功能器件的量子精度多尺度建模奠定了基础。