Machine-learning (ML) force fields enable large-scale simulations with near-first-principles accuracy at substantially reduced computational cost. Recent work has extended ML force-field approaches to adiabatic dynamical simulations of condensed-matter lattice models with coupled electronic and structural or magnetic degrees of freedom. However, most existing formulations rely on hand-crafted, symmetry-aware descriptors, whose construction is often system-specific and can hinder generality and transferability across different lattice Hamiltonians. Here we introduce a symmetry-preserving framework based on equivariant neural networks (ENNs) that provides a general, data-driven mapping from local configurations of dynamical variables to the associated on-site forces in a lattice Hamiltonian. In contrast to ENN architectures developed for molecular systems -- where continuous Euclidean symmetries dominate -- our approach aims to embed the discrete point-group and internal symmetries intrinsic to lattice models directly into the neural-network representation of the force field. As a proof of principle, we construct an ENN-based force-field model for the adiabatic dynamics of the Holstein Hamiltonian on a square lattice, a canonical system for electron-lattice physics. The resulting ML-enabled large-scale dynamical simulations faithfully capture mesoscale evolution of the symmetry-breaking phase, illustrating the utility of lattice-equivariant architectures for linking microscopic electronic processes to emergent dynamical behavior in condensed-matter lattice systems.

翻译：机器学习力场能够以接近第一性原理的精度进行大规模模拟，同时显著降低计算成本。近期研究已将机器学习力场方法扩展至具有耦合电子与结构（或磁性）自由度的凝聚态晶格模型的绝热动力学模拟。然而，现有大多数方法依赖于人工构建的对称性感知描述符，其构造通常具有体系特异性，可能阻碍不同晶格哈密顿量间的通用性与可迁移性。本文提出一种基于等变神经网络的对称性保持框架，该框架提供了从动力学变量的局域构型到晶格哈密顿量中相应格点力的通用数据驱动映射。与针对分子体系（其对称性以连续欧几里得对称性为主）开发的等变神经网络架构不同，我们的方法旨在将晶格模型固有的离散点群对称性与内禀对称性直接嵌入力场的神经网络表示中。作为原理验证，我们为方晶格上霍尔斯塔姆哈密顿量的绝热动力学构建了基于等变神经网络的力场模型，该模型是电子-晶格物理的典型体系。由此实现的机器学习驱动的大规模动力学模拟，精确捕捉了对称破缺相的介观演化过程，证明了晶格等变架构在连接凝聚态晶格系统中微观电子过程与涌现动力学行为方面的实用性。