GMT: A Geometric Multigrid Transformer Solver for Microstructure Homogenization

TL;DR

GMT is a neural solver that combines geometric multigrid principles with transformer architecture to efficiently and accurately perform microstructure homogenization, significantly speeding up computations while maintaining high fidelity.

Contribution

The paper introduces GMT, a novel neural multigrid transformer that enforces physical consistency and achieves high accuracy and speed in lattice homogenization tasks.

Findings

Achieves 10^{-5} residual errors with 160x speedup over GPU solvers.

Generalizes well to unseen geometries and non-periodic conditions.

Scales efficiently to high resolutions like 512^3.

Abstract

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.