Asymptotic Theory of $ell_1$-Regularized PDE Identification from a Single Noisy Trajectory

We prove the support recovery for a general class of linear and nonlinear evolutionary partial differential equation (PDE) identification from a single noisy trajectory using $ell_1$ regularized Pseudo-Least Squares model~($ell_1$-PsLS). In any associative $mathbb{R}$-algebra generated by finitely many differentiation operators that contain the unknown PDE operator, applying $ell_1$-PsLS to a given data set yields a family of candidate models with coefficients $mathbf{c}(lambda)$ parameterized by the regularization weight $lambdageq 0$. The trace of ${mathbf{c}(lambda)}_{lambdageq 0}$ suffers from high variance due to data noises and finite difference approximation errors. We provide a set of sufficient conditions which guarantee that, from a single trajectory data denoised by a Local-Polynomial filter, the support of $mathbf{c}(lambda)$ asymptotically converges to the true signed-support associated with the underlying PDE for sufficiently many data and a certain range of $lambda$. We also show various numerical experiments to validate our theory.