A Multigrid-inspired Neural Iterative Solver for Poisson Equations on Large Voxel Grids

Kangbo Lyu, Ruihong Cen, Yushen Wu, Tao Du

Keywords: numerical method; neural preconditioner; Poisson equation;

TL;DR: Our neural PCG use 100~200ms for accurately solving 16M-DoF linear systems, 5-10x faster than previous neural methods

Abstract: Solving the Poisson equation at scale remains a critical computational bottleneck, notably in projection-based fluid simulation. The multigrid-preconditioned conjugate gradient (MGPCG) method is the widely recognized state-of-the-art (SOTA) solver for such problems, featuring parallelized implementations and fast convergence rates. However, it does not leverage data priors and requires domain-specific effort to tune. Recent developments in neural iterative solvers explore the possibility of learning solvers from data, yet they fall short in terms of time efficiency and scalability compared to multigrid solvers tailored to specific problems. Building on advances in neural preconditioners and multigrid methods, we present a hybrid dual-channel neural multigrid preconditioner that incorporates a classical smoothing channel and a novel neural convolutional channel designed as a learnable non-linear mapping with respect to the matrix. The careful design combines the high-performance of multigrid frameworks with the adaptability and generalizability of learned models, while ensuring the symmetric positive definite property of the preconditioner. Evaluated on a new, large-scale fluid dataset with more than 6,000 instances and 16-64M unknowns each, our method outperforms prior neural preconditioners by 5-10$\times$ and, for the first time, achieves a 1.03–1.26$\times$ speedup over a highly tuned geometric multigrid solver.

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Submission Number: 82