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

The identification of Partial Differential Equations (PDEs) has emerged as a prominent data-driven approach for mathematical modeling and has attracted considerable attention in recent years. 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 \textbf{W}eak formulation of \textbf{G}roup-sparsity-based framework for \textbf{IDENT}ifying PDEs with varying coefficients, called \textbf{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.