使用采样局部弱识别（Sampled Local WeakIdent，SLW-Ident）识别变化的偏微分方程
Identifying changing partial differential equations using Sampled Local WeakIdent
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中文总结 AI 辅助
该研究提出SLW-Ident框架，通过采样斑块、结合残差误差与全局信息，可从含噪数据中准确识别变化偏微分方程的区域及控制方程，解决了Local WeakIdent对局部扰动敏感的问题。
中文摘要 AI 辅助
我们提出了Sampled Local WeakIdent（SLW-Ident），这是一个用于从给定的单一数据集识别变化控制方程的框架。与使用基于有限元的近似来表示变化系数的典型方法不同，本文探索了一种用于识别微分方程的局部方法。首先，我们展示了Local WeakIdent的优势，它能实现良好的局部识别，且在小斑块尺寸下计算效率高，但它对局部扰动较为敏感。我们提出的SLW-Ident可稳定识别过程，还整合了全局信息：我们首先在整个给定域中采样斑块，为每个采样斑块识别方程，然后利用这些方程的残差误差来寻找不同方程之间的过渡。我们将已识别方程的支撑集不发生变化的区域称为单方程区域。在每个单方程区域内，我们选择出现频率最高的方程作为已识别方程，并在每个单方程区域内找到常数及变系数偏微分方程（PDE）。这一方法得到了不确定性量化理论的支撑，该理论给出了主导支撑选择的统计误差与斑块数量之间的关系。我们开展了各类数值实验，结果表明，即便在给定数据存在噪声的情况下，SLW-Ident也能准确恢复单方程区域以及变系数变化PDE的控制方程。
英文摘要
We propose Sampled Local WeakIdent (SLW-Ident), a framework for identifying changing governing equations from a single set of given data. Different from a typical approach of using finite element based approximation to represent varying coefficients, this paper explores a local approach in identification of differential equations. First, we present the power of Local WeakIdent which gives good local identification and is also computationally efficient with a small patch size, yet it can be sensitive to local perturbations. We propose SLW-Ident which stabilizes the identification process and also incorporates global information: we first sample patches in the whole given domain, identify equations for each sampled patch, then use residual error of these equations to find the transitions between different equations. We refer to a region where the support of the identified equation does not change to be a region of one equation. Within each region of one equation, we pick the most frequently identified equation as the identified equation, and find constant as well as varying coefficient PDEs within each region of one equation. This is justified by an uncertainty quantification theory that gives the relation between statistical error of dominant support selection and the number of patches. We provide various numerical experiments showing that SLW-Ident accurately recovers the regions of one equation and the governing equations for changing PDEs with varying coefficients even with noisy given data.
