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This research presents a novel AI-driven preconditioner using graph-and-attention networks to solve sparse linear systems for real-time interactive simulation within strict 8–16 ms budgets. By optimizing the training objective with a truncated Kaporin condition number, the method significantly reduces conjugate-gradient iterations compared to traditional approaches like algebraic multigrid or Jacobi. The technique enables high-fidelity physics simulations on complex 3D meshes by leveraging H²-matrix formats to maintain linear computational complexity relative to problem size.
A Neural Hierarchical-Matrix Preconditioner for Real-Time GPU Solves
arXiv:2605.13343v3 Announce Type: replace-cross 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: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.
Accepted to SIGGRAPH Asia 2026 Posters (SA Posters '26). 3 pages; 2-page supplement as ancillary file. Supersedes v1-v2 (Hierarchical Transformer Preconditioning for Interactive Physics Simulation): their cosine-Hutchinson objective carries a factor lambda in its gradient on near-null modes; this version trains a truncated Kaporin condition number and evaluates on 3D tet meshes
Graphics (cs.GR); Distributed, Parallel, and Cluster Computing (cs.DC); Machine Learning (cs.LG); Numerical Analysis (math.NA)
arXiv:2605.13343 [cs.GR]
(or arXiv:2605.13343v3 [cs.GR] for this version)
https://doi.org/10.1145/3829333.3847989 DOI(s) linking to related resources
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