Lattice metamaterials enable lightweight, multifunctional structures, yet homogenization-based evaluation of their effective properties remains computationally expensive. Neural surrogates offer speed but often lack the accuracy and stability required for engineering-grade simulations. We introduce GMT, a Geometric Multigrid Transformer -- a neural solver with high numerical fidelity for fast and reliable lattice homogenization. GMT achieves architectural alignment with Geometric Multigrid (GMG) by restructuring Point Transformer V3 to operate across sparse GMG hierarchies, capturing long-range dependencies and cross-level interactions essential for multigrid convergence. To enforce physical consistency, GMT incorporates physics-aware positional encoding for strict enforcement of periodicity and predicts both the finest-level solution and multi-level residual corrections. These predictions deliver a spectrally-aligned initialization, enabling end-to-end training under physics-informed and solver-aware losses and requiring only a single GMG V-cycle refinement to reach convergence. This fusion of neural prediction and numerical rigor achieves relative residual errors of $10^{-5}$ with a $160\times$ speedup over state-of-the-art GPU-based solvers at equivalent accuracy -- particularly at high resolutions (e.g. $512^3$), where traditional methods become most costly. We validate GMT across mechanical and thermal domains, demonstrate robust generalization to unseen geometries and non-periodic settings, and showcase scalability to high resolutions -- enabling real-time design iteration, multi-scale simulations, high-throughput material discovery, and inverse design.

翻译：晶格超材料能够实现轻量化的多功能结构，但基于均质化方法评估其有效性能仍存在计算成本高昂的问题。神经网络代理模型虽能提升计算速度，但往往缺乏工程级仿真所需的精度与稳定性。我们提出GMT（几何多重网格Transformer）——一种兼具高数值保真度的神经求解器，可快速可靠地实现晶格均质化。通过重构Point Transformer V3以适配稀疏几何多重网格层级架构，GMT实现了与几何多重网格（GMG）的架构对齐，捕获了对多重网格收敛至关重要的长程依赖与跨层级交互。为强制执行物理一致性，GMT采用物理感知位置编码严格保障周期性条件，并同时预测最细网格层级解与多层级残差修正项。这些预测提供频谱对齐的初始化解，支持在物理信息引导与求解器感知损失函数下的端到端训练，仅需单次GMG V循环精炼即可达到收敛。这种神经预测与数值严谨性的融合实现了$10^{-5}$的相对残差误差，在等效精度下相比基于GPU的最先进求解器获得160倍加速——尤其在高分辨率（如$512^3$）场景中，传统方法计算成本最为高昂。我们通过力学与热学领域验证GMT性能，证明其对未见几何构型与非周期设置的鲁棒泛化能力，并展示其在高分辨率下的可扩展性——可支持实时设计迭代、多尺度仿真、高通量材料发现及逆向设计。