AI 中文总结
研究针对参数动力系统中局部解流形跟踪问题，提出结合局部POD降维等多种技术的残差认证自适应方法，经多模型数值研究验证，该方法能集中高保真计算于困难参数区，保持相关联系。
AI 中文摘要
本文提出了一种用于跟踪参数动力系统中局部解流形的残差认证自适应方法。该方法结合了局部本征正交分解降维、全物理残差检查、状态距离快照遗忘、高保真重采样以及轻量级物理信息神经校正。算法不学习一个全局参数到状态的映射，而是维护当前活跃的局部分支，并在残差表明有效性丧失时进行更新。分析解释了为何通过局部残差误差控制残差阈值在正则分支上有意义，以及为何在折叠或其他退化邻域附近需要更严格的局部更新。对奥斯特瓦尔德熟化、粒子数平衡模型和布拉图方程的数值研究在低维动力学、非线性非局部残差补偿和近折叠模型失效等方面测试了该方法。结果表明，残差认证的局部模型管理可以将高保真计算集中在困难的参数区域，同时保持代理预测、物理一致性和活跃分支跟踪之间的可解释联系。
英文摘要
This paper presents a residual-certified adaptive method for tracking local solution manifolds in parametric dynamical systems. The method combines local POD reduction, full physical residual checks, state-distance snapshot forgetting, high-fidelity resampling, and a lightweight physics-informed neural correction. Instead of learning one global parameter-to-state map, the algorithm maintains the currently active local branch and updates it when the residual indicates loss of validity. The analysis explains why residual thresholds are meaningful on regular branches through local residual-error control, and why stricter local updates are needed near folds or other degenerate neighborhoods. Numerical studies on Ostwald ripening, a particle population-balance model, and the Bratu equation test the approach across low-dimensional dynamics, nonlinear nonlocal residual compensation, and near-fold model failure. The results show that residual-certified local model management can concentrate high-fidelity computation in difficult parameter regions while preserving an interpretable link between surrogate prediction, physical consistency, and active-branch tracking.
Comments32 pages, 4 figures
