{
  "abstract": "Interactive simulation solves Ax=b for a sparse SPD A that changes every frame, inside an 8-16 ms budget. At a few thousand unknowns, the setup of algebraic multigrid alone exceeds that budget, while Jacobi and other local preconditioners have no setup but cannot move error across the domain. We learn a preconditioner for this gap: a graph-and-attention network predicts an SPD approximate inverse in H^2-matrix format. On a spatially ordered 3D mesh, blocks of the true inverse lose rank as the clusters they couple move apart; the nested bases of the format follow that decay, so inference and apply are dominated by leaf-block work linear in N, where a dense inverse costs N^2. Our main finding concerns training. Probe losses reach M only through a product with A, so their gradient vanishes on the near-null modes that set the conjugate-gradient iteration count. A truncated Kaporin condition number has no such factor; changing only the objective cuts iterations on a held-out frame from 116 to 33. On a ladder of stiff tetrahedral diffusion problems ours alone fits an 8.3 ms (120 fps) frame from N=572 to 3,647.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.13343",
  "author_total": 4,
  "authors": [
    {
      "name": "Carl Osborne"
    },
    {
      "name": "Minghao Guo"
    },
    {
      "name": "Crystal Owens"
    },
    {
      "name": "Wojciech Matusik"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bgg6c23ny444x.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.13343",
  "fulltext_url": null,
  "id": "bgg6c23ny444x",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.13343",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.DC"
    ],
    [
      "arxiv",
      "cs.GR"
    ],
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "openalex",
      "1703|Matrix Theory and Algorithms"
    ]
  ],
  "openalex": "W7161149378",
  "pdf_url": "https://arxiv.org/pdf/2605.13343",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bgg6c23ny444x.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2605.13343",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.13343"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "A Neural Hierarchical-Matrix Preconditioner for Real-Time GPU Solves",
  "title_text": "A Neural Hierarchical-Matrix Preconditioner for Real-Time GPU Solves",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/bgg6c23ny444x",
  "venue": "arXiv",
  "versions": [
    {
      "doi": "10.1145/3829333.3847989",
      "pid": null,
      "relation": "published-as",
      "title": null,
      "venue": null
    }
  ]
}