AI 中文总结
本文针对新的非线性模型降阶方法PPMD开展收敛性分析，通过耦合扰动分析推导了其轨迹误差界与概率一致性，明确了限制模型精度的误差组件。
AI 中文摘要
本文对一种新开发的非线性模型降阶方法——参数概率流形分解（parametric probabilistic manifold decomposition，PPMD）进行收敛性分析。此外，现有非线性降阶模型的分析通常将子空间降阶、流形表示、回归和非线性重构视为独立组件，且往往仅停留在离散状态向量层面。据我们所知，目前尚无理论能追踪这种依赖数据的模型中的完整误差传播，该模型的基、残差几何、谱坐标、参数映射和提升算子均从同一数值解数据中学习得到。我们针对整个PPMD过程开展耦合扰动分析：空间离散化与时间求积诱导的轨迹几何将离散轨迹向量与对应PDE范数等距连接；引入总体谱对象以对齐经验残差坐标并推导均匀坐标误差估计，随后量化其通过希尔伯特值核提升估计器的传播；将这些结果与全阶离散化误差、加权低秩近似、参数回归及残差表示缺陷相结合，得到确定性和高概率的轨迹误差界，以及连续PDE轨迹空间中的概率一致性。该理论明确了主要误差如何相互作用，以及哪些组件限制了非线性降阶模型的精度。
英文摘要
This paper presents a convergence analysis for a newly developed nonlinear model reduction method: parametric probabilistic manifold decomposition (PPMD)~\cite{guo2026parametric}. In addition, existing analyzes of nonlinear reduced order models typically treat subspace reduction, manifold representation, regression, and nonlinear reconstruction as separate components and often remain at the level of discrete state vectors. To the best of our knowledge, no theory tracks the complete error propagation in a data-dependent model whose basis, residual geometry, spectral coordinates, parameter maps, and lifting operator are all learned from the same numerical solution data. We develop a coupled perturbation analysis for the entire PPMD procedure. A trajectory geometry induced by the spatial discretization and temporal quadrature connects discrete trajectory vectors isometrically with the corresponding PDE norm. Population spectral objects are introduced to align the empirical residual coordinates and derive a uniform coordinate error estimate, whose propagation through the Hilbert-valued kernel lifting estimator is then quantified. Combining these results with the full order discretization error, weighted low-rank approximation, parameter regression, and residual representation defect yields deterministic and high-probability trajectory error bounds and consistency in probability in the continuous PDE trajectory space. The theory identifies how the principal errors interact and which components limit the accuracy of the nonlinear reduced order model.
