GMT: A Geometric Multigrid Transformer Solver for Microstructure Homogenization

2026-04-29

AI Summary

This work addresses the high computational cost of homogenization in conventional lattice metamaterials and the difficulty of existing neural surrogates in simultaneously achieving accuracy and stability. The authors propose a novel neural solver that integrates geometric multigrid (GMG) with Point Transformer V3, reformulating it for the first time as a cross-scale architecture within a sparse GMG hierarchy. Key innovations include physics-aware positional encoding, spectral alignment initialization, multi-level residual correction, and a physics-informed loss function enabling end-to-end training. The method achieves a 160× acceleration at 512³ resolution with relative residual errors as low as 10⁻⁵, supports thermo-mechanical multiphysics, generalizes to unseen geometries and non-periodic structures, and is suitable for real-time simulation and inverse design.

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.

Problem

• homogenization
• lattice metamaterials
• computational efficiency
• neural surrogates
• numerical accuracy

Innovation

• Geometric Multigrid Transformer
• neural solver
• physics-informed learning
• lattice homogenization
• spectrally-aligned initialization