用低秩谱表示替代图聚合，提升偏微分方程求解速度
NeuraLSP: A Neural Spectral Preconditioner for Accelerating PDE Solvers
- 用左奇异子空间构建固定低秩谱表示，取代传统图聚类
- 在多种偏微分方程上实现最高53%的求解加速
- 适合需要高效线性系统求解的科学计算研究者
大规模稀疏线性系统源于偏微分方程（PDE）求解，是高性能科学计算的核心问题，预条件器至关重要。多重网格方法是最有效的预条件器之一，但其性能依赖于网格转移算子的精确构造。当前神经多重网格方法使用图神经网络（GNN）从离散化系统矩阵中提取连通性来学习这些算子。尽管有效，这类基于图的方法存在秩膨胀问题，导致粗网格空间过大，收敛变慢。本文提出NeuraLSP，一种新型神经多重网格预条件器，将图聚合替换为来自近零空间成分左奇异子空间的固定低秩谱表示。在网络设计层面，采用新型子空间损失函数训练，保留对多重网格收敛最相关的误差模式，同时抑制秩膨胀。该工作兼具理论保证与对秩膨胀的实证鲁棒性，在多种PDE族上相较现有最先进神经预条件器实现高达53%的加速。
原文摘要 · Abstract (English)
Solving large-scale sparse linear systems originating from partial differential equations (PDEs) is a fundamental topic in high-performance scientific computing, where preconditioners are crucial. Multigrid methods are among the most effective preconditioners, yet their performance is dictated by the accurate construction of grid transfer operators. Current neural multigrid methods learn such operators with graph neural networks (GNNs), typically by extracting connectivity from discretized system matrices. While effective, these graph-based constructions suffer from rank inflation, resulting in unnecessarily large coarse spaces and slower convergence. To ameliorate, this paper advocates NeuraLSP, a new neural multigrid preconditioner that replaces graph aggregation with a fixed low-rank spectral representation derived from the left singular subspace of near-nullspace components. At the network design level, NeuraLSP is trained with a novel subspace loss function, which preserves the error modes most relevant to multigrid convergence while suppressing rank inflation. This paper's grand innovation hinges upon both theoretical guarantees and empirical robustness to rank inflation, affording up to a 53% speedup over SOTA neural preconditioners across a variety of PDE families.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。