This study presents an approach to discover the significant terms in partial differential equations (PDEs) that describe particular phenomena based on the measured data. The relationship between the known field data and its continuous representation of PDEs is achieved through a linear regression model. It specifically employs the peridynamic differential operator (PDDO) and sparse linear regression learning algorithm. The PDEs are approximated by constructing a feature matrix, velocity vector and unknown coefficient vector. Each candidate term (derivatives) appearing in the feature matrix is evaluated numerically by using the PDDO. The solution to the regression model with regularization is achieved through Douglas-Rachford (D-R) algorithm which is based on proximal operators. This coupling performs well due to their robustness to noisy data and the calculation of accurate derivatives. Its effectiveness is demonstrated by considering several fabricated data associated with challenging nonlinear PDEs such as Burgers, Swift-Hohenberg (S-H), Korteweg-de Vries (KdV), Kuramoto-Sivashinsky (K-S), nonlinear Schrödinger (NLS) and Cahn-Hilliard (C-H) equations.

这项研究提出了一种发现偏微分方程（PDE）中重要项的方法，这些项根据测量数据描述了特定现象。通过线性回归模型可以实现已知现场数据与其连续表示的PDE之间的关系。它特别采用了绕动微分算子（PDDO）和稀疏线性回归学习算法。通过构造特征矩阵，速度向量和未知系数向量来近似PDE。通过使用PDDO对出现在特征矩阵中的每个候选项（导数）进行数值评估。通过基于近端算子的Douglas-Rachford（DR）算法，可以对带有正则化的回归模型进行求解。由于它们对嘈杂数据的鲁棒性和精确导数的计算，因此这种耦合表现良好。通过考虑与挑战性非线性PDE相关的一些伪造数据来证明其有效性，例如Burgers，Swift-Hohenberg（SH），Korteweg-de Vries（KdV），Kuramoto-Sivashinsky（KS），非线性Schrödinger（NLS）和Cahn-Hilliard（ CH）方程。