Asymptotic Theory of `₁-Regularized PDE Identification from a Single Noisy Trajectory

We provide a formal theoretical analysis on the PDE identification via the `1-regularized pseudo least square method from the statistical point of view. In this article, we assume that the differential equation governing the dynamic system can be represented as a linear combination of various linear and nonlinear differential terms. Under noisy observations, we employ local-polynomial fitting for estimating state variables and apply the `1 penalty for model selection. Our theory proves that the classical mutual incoherence condition on the feature matrix F and the β^∗_min-condition for the ground-truth signal β^∗ are sufficient for the signed-support recovery of the `1-PsLS method. We run numerical experiments on two popular PDE models, the viscous Burgers and the Korteweg-de Vries (KdV) equations, and the results from the experiments corroborate our theoretical predictions.