Identification of Partial-Differential-Equations-Based Models from Noisy Data via Splines

We propose a two-stage method called $\mathit{\text{Spline}}\mathit{\text{Assisted}}\mathit{\text{Partial}}\mathit{\text{Differential}}\mathit{\text{Equation}}\mathit{\text{based}}\mathit{\text{Model}}\mathit{\text{Identification}}\mathit{\text{(SAPDEMI)}}$ 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.