  In this paper, we propose a two-stage method called Spline Assisted Partial
Differential Equation involved Model Identification (SAPDEMI) to efficiently
identify the underlying partial differential equation (PDE) models from the
noisy data. In the first stage -- functional estimation stage -- we employ the
cubic spline to estimate the unobservable derivatives, which serve as
candidates included the underlying PDE models. The contribution of this stage
is that, it is computational efficient because it only requires the
computational complexity of the linear polynomial of the sample size, which
achieves the lowest possible order of complexity. In the second stage -- model
identification stage -- we apply Least Absolute Shrinkage and Selection
Operator (Lasso) to identify the underlying PDE models. The contribution of
this stage is that, we focus on the model selections, while the existing
literature mostly focuses on parameter estimations. Moreover, we develop
statistical properties of our method for correct identification, where the main
tool we use is the primal-dual witness (PDW) method. Finally, we validate our
theory through various numerical examples.
