GMT: A Geometric Multigrid Transformer Solver for Microstructure Homogenization
For engineers and researchers performing microstructure homogenization, GMT provides a fast and accurate alternative to traditional solvers, enabling real-time design and multi-scale simulations.
GMT introduces a neural solver that combines a Geometric Multigrid Transformer with physics-informed losses to achieve relative residual errors of 10^{-5} and a 160x speedup over GPU-based solvers for lattice homogenization, particularly at high resolutions like 512^3.
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.