AI 中文总结
针对非线性消除预处理中识别收敛慢分量的挑战，提出在线学习增强的NE预条件器，从残差主导结构识别不良子集，集成到并行区域分解框架，实验表明该方法能产生可靠连贯的不良子集，收敛性优于基线。
AI 中文摘要
非线性预处理不精确牛顿法是求解由偏微分方程离散产生的大规模非线性代数系统的有效方法。非线性消除（NE）预处理的一个核心挑战是可靠识别要消除的收敛缓慢的分量。现有选择策略常依赖特定问题物理信息或直接应用于原始非线性残差的用户调整阈值，这在停滞区域附近可能包含不规则振荡结构，使所选不良子集对阈值参数高度敏感。在这项工作中，我们提出一种在线学习增强的NE预条件器，它从非线性残差的主导结构而非原始残差本身识别不良子集。在当前牛顿求解的停滞阶段在线收集残差快照，并训练无监督提取模型以捕获主要非线性不平衡。我们考虑基于主成分分析的线性提取器和基于自动编码器神经网络的非线性提取器。此外，我们将该方法集成到并行区域分解框架中，在每个子域上独立训练局部提取模型。然后使用学习到的残差重构来定义不良子集并指导非线性消除过程。在雷诺数高达10000的顶盖驱动腔流上的数值实验表明，该方法产生更可靠和连贯的不良子集，对NE和学习参数都具有鲁棒性，并且在收敛方面优于基线NE预条件器。
英文摘要
Nonlinearly preconditioned inexact Newton methods form an effective class of solvers for large-scale nonlinear algebraic systems arising from the discretization of partial differential equations. A central challenge in nonlinear elimination (NE) preconditioning is the reliable identification of the slowly converging components to be eliminated. Existing selection strategies often rely on problem-specific physical information or user-tuned thresholds applied directly to the raw nonlinear residual, which may contain irregular oscillatory structures near stagnation regions, making the selected bad subset highly sensitive to threshold parameters. In this work, we propose an online-learning-enhanced NE preconditioner that identifies the bad subset from the dominant structure of the nonlinear residual rather than from the raw residual itself. Residual snapshots are collected online during the stagnation phase of the current Newton solve, and an unsupervised extraction model is trained to capture the principal nonlinear imbalance. We consider both a linear extractor based on principal component analysis and nonlinear extractors based on autoencoder neural networks. Moreover, we integrate the approach into a parallel domain decomposition framework, which trains a local extraction model independently on each subdomain. The learned residual reconstruction is then used to define the bad subset and guide the nonlinear elimination process. Numerical experiments on lid-driven cavity flows at Reynolds numbers up to 10,000 show that the proposed method produces more reliable and coherent bad subsets, is robust with respect to both NE and learning parameters, and outperforms the baseline NE preconditioner in terms of the convergence.
