Title: Closing the Performance Gap in Neural Conjugate Gradient Method: A Hybrid Multigrid Preconditioning Approach
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)
Keywords: numerical method; neural preconditioner; partial differential equation;

Abstract: Recent studies on neural preconditioners for the conjugate gradient (CG) methods have shown promise but sometimes over-optimistic: methods targeting small-scale problems are compared with decomposition-based preconditioners but do not scale; those aimed at moderate- to large-scale problems are typically 5$\times$ slower in wall-clock time than state-of-the-art multigrid (MG) preconditioners, with worse iteration counts and higher model complexity. To address this gap, we revisit the designs of neural preconditioners and identify the key trade-off: methods tailored for scalability lack the expressiveness to emulate effective smoothers, whereas highly expressive designs struggle to scale. Building on these insights, we introduce a dual-channel neural multigrid preconditioner that couples a classical smoothing path with a lightweight neural convolutional path. This architecture preserves the minimal expressiveness and symmetric positive definite property while injecting data-driven adaptability. Our method demonstrates, for the first time, that neural preconditioners can surpass SOTA MG preconditioners on large-scale problems, achieving a 1.03-1.26$\times$ speedup on Poisson equations and 2-3$\times$ acceleration on other second-order PDEs involving up to 64 million unknowns, while also delivering 5-10$\times$ improvements over existing neural methods. These results establish a new benchmark for neural preconditioning.

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