RINS：面向大型稀疏线性系统的残差图像神经子空间求解器

中文摘要 AI 辅助

由偏微分方程离散化产生的大型稀疏线性系统需要校正子空间，其算子图像能够解释当前残差。我们研究了这一残差图像视角，并提出了Gate-RINS，一种神经子空间求解器，它从缓存的残差探针生成多项式校正基，并通过轻量级的、依赖于残差和坐标的逐点门控对其进行调制。投影最小二乘更新保持不变，因此神经组件仅选择展开方向，而数值闭包通过\\(\operatorname{range}(AQ_t)\\)对它们进行测试。我们还引入了一种混合控制器调度，该调度在同一投影求解器下将GRANS风格的图控制器与Gate-RINS组合在一起。在六个源自PDE的基准任务和两种规模上，Gate-RINS在大多数设置中达到固定相对残差阈值的同步墙钟时间快于GMRES和最近的纯图神经基线，并且混合调度进一步改善了残差轨迹。困难模式诊断和轨迹可视化支持了这些增益与算子图像子空间更有效地与当前残差对齐相关的解释。

英文摘要

Large sparse linear systems from PDE discretizations require correction subspaces whose operator images explain the current residual. We study this residual-image viewpoint and propose Gate-RINS, a neural subspace solver that generates polynomial correction bases from cached residual probes and modulates them with a lightweight residual- and coordinate-dependent pointwise gate. The projected least-squares update remains unchanged, so the neural component only chooses the expansion directions while the numerical closure tests them through \(\operatorname{range}(AQ_t)\). We also introduce a hybrid controller schedule that composes GRANS-style graph controllers with Gate-RINS under the same projected solver. Across six PDE-derived benchmark tasks and two scales, Gate-RINS reaches fixed relative-residual thresholds faster in synchronized wall-clock time than GMRES and a recent graph-only neural baseline in most settings, and hybrid schedules further improve the residual trajectory. Difficult-mode diagnostics and trajectory visualizations support the interpretation that these gains are associated with operator-image subspaces that align more effectively with the current residual.

发表机构

• Shanghai Innovation Institute（上海创新研究院）
• Harbin Institute of Technology（哈尔滨工业大学）
• Shanghai Jiao Tong University（上海交通大学）
• Renmin University of China（中国人民大学）

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