We propose a two-stage method called \textit{Spline Assisted Partial Differential Equation based Model Identification (SAPDEMI)} to identify partial differential equation (PDE)-based models from noisy data. In the first stage, we employ the cubic splines to estimate unobservable derivatives. The underlying PDE is based on a subset of these derivatives. This stage is computationally efficient: its computational complexity is a product of a constant with the sample size; this is the lowest possible order of computational complexity. In the second stage, we apply the Least Absolute Shrinkage and Selection Operator (Lasso) to identify the underlying PDE-based model. Statistical properties are developed, including the model identification accuracy. We validate our theory through various numerical examples and a real data case study. The case study is based on a National Aeronautics and Space Administration (NASA) data set.

翻译：我们提出了一种名为“样条辅助偏微分方程模型辨识（SAPDEMI）”的两阶段方法，用于从含噪数据中辨识基于偏微分方程（PDE）的模型。在第一阶段，我们采用三次样条函数估计不可观测的导数，基础偏微分方程由这些导数的子集构成。该阶段计算效率高：其计算复杂度为常数与样本容量的乘积，属于最低阶计算复杂度。在第二阶段，我们应用最小绝对收缩与选择算子（Lasso）来辨识基于偏微分方程的模型。我们推导了包括模型辨识精度在内的统计性质，并通过多种数值算例及一项基于美国国家航空航天局（NASA）数据集的真实案例验证了理论结果。