标题： WG-IDENT：具有变系数的偏微分方程的弱群识别 显示英文标题

标题： WG-IDENT: Weak Group Identification of PDEs with Varying Coefficients

摘要： 偏微分方程（PDEs）的识别是一种用于数学建模的数据驱动方法，并且最近引起了大量关注。从高度噪声的空间时间数据中识别PDE的稳定性和精度提出了显著的困难。当PDE的系数受空间变化影响时，这个问题变得更加复杂。在本文中，我们提出了一种基于群稀疏性的弱形式框架来识别具有变化系数的PDEs，称为WG-IDENT，以应对这一挑战。我们的方法利用PDE的弱形式来减少噪声的影响。我们使用B样条表示测试函数和未知的PDE系数，其中测试函数的节点向量是根据噪声数据的谱分析最优选择的。为了促进特征选择，我们提出将群稀疏回归与一种新设计的群特征修剪技术GF-trim相结合，以消除不重要的特征。进行了广泛的对比消融研究以验证我们提出的方法。所提出的方法不仅展示了比最先进的算法更高的抗高噪声水平的能力，而且在表现出对超参数选择的降低敏感性的同时实现了优越的性能。 显示英文摘要

摘要： Partial Differential Equations (PDEs) identification is a data-driven method for mathematical modeling, and has received a lot of attentions recently. The stability and precision in identifying PDE from heavily noisy spatiotemporal data present significant difficulties. This problem becomes even more complex when the coefficients of the PDEs are subject to spatial variation. In this paper, we propose a Weak formulation of Group-sparsity-based framework for IDENTifying PDEs with varying coefficients, called WG-IDENT, to tackle this challenge. Our approach utilizes the weak formulation of PDEs to reduce the impact of noise. We represent test functions and unknown PDE coefficients using B-splines, where the knot vectors of test functions are optimally selected based on spectral analysis of the noisy data. To facilitate feature selection, we propose to integrate group sparse regression with a newly designed group feature trimming technique, called GF-trim, to eliminate unimportant features. Extensive and comparative ablation studies are conducted to validate our proposed method. The proposed method not only demonstrates greater robustness to high noise levels compared to state-of-the-art algorithms but also achieves superior performance while exhibiting reduced sensitivity to hyperparameter selection.