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Graph."],"published-print":{"date-parts":[[2026,7,3]]},"abstract":"<jats:p>\n                    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\n                    <jats:italic toggle=\"yes\">G<\/jats:italic>\n                    eometric\n                    <jats:italic toggle=\"yes\">M<\/jats:italic>\n                    ultigrid\n                    <jats:italic toggle=\"yes\">T<\/jats:italic>\n                    ransformer - a neural solver with high numerical fidelity for fast and reliable lattice homogenization. GMT achieves\n                    <jats:italic toggle=\"yes\">architectural alignment<\/jats:italic>\n                    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\n                    <jats:italic toggle=\"yes\">spectrally-aligned initialization<\/jats:italic>\n                    , 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\n                    <jats:sup>-5<\/jats:sup>\n                    with a 160\u00d7 speedup over state-of-the-art GPU-based solvers at equivalent accuracy - particularly at high resolutions\n                    <jats:italic toggle=\"yes\">(e.g.<\/jats:italic>\n                    , 512\n                    <jats:sup>3<\/jats:sup>\n                    ), 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.\n                  <\/jats:p>","DOI":"10.1145\/3811333","type":"journal-article","created":{"date-parts":[[2026,7,3]],"date-time":"2026-07-03T07:05:51Z","timestamp":1783062351000},"page":"1-15","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":0,"title":["GMT: A Geometric Multigrid Transformer Solver for Microstructure Homogenization"],"prefix":"10.1145","volume":"45","author":[{"ORCID":"https:\/\/orcid.org\/0009-0009-1522-6145","authenticated-orcid":false,"given":"Yu","family":"Xing","sequence":"first","affiliation":[{"name":"Shandong University, Qingdao, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3768-6654","authenticated-orcid":false,"given":"Yang","family":"Liu","sequence":"additional","affiliation":[{"name":"Microsoft Research Asia, Beijing, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1400-3566","authenticated-orcid":false,"given":"Tianyang","family":"Xue","sequence":"additional","affiliation":[{"name":"The University of Hong Kong, Hong Kong, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5881-892X","authenticated-orcid":false,"given":"Lin","family":"Lu","sequence":"additional","affiliation":[{"name":"Shandong University, Qingdao, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"320","published-online":{"date-parts":[[2026,7,3]]},"reference":[{"key":"e_1_2_2_1_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.addma.2020.101562"},{"key":"e_1_2_2_2_1","doi-asserted-by":"publisher","DOI":"10.1137\/21M1433514"},{"key":"e_1_2_2_3_1","volume-title":"International Conference on Learning Representations.","author":"Brandstetter Johannes","year":"2022","unstructured":"Johannes Brandstetter, Daniel E. 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