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We prove the support recovery for a general class of linear and nonlinearevolutionary partial differential equation (PDE) identification from a singlenoisy trajectory using $\ell_1$ regularized Pseudo-Least Squaresmodel~($\ell_1$-PsLS). In any associative $\mathbb{R}$-algebra generated byfinitely many differentiation operators that contain the unknown PDE operator,applying $\ell_1$-PsLS to a given data set yields a family of candidate modelswith coefficients $\mathbf{c}(\lambda)$ parameterized by the regularizationweight $\lambda\geq 0$. The trace of $\{\mathbf{c}(\lambda)\}_{\lambda\geq 0}$suffers from high variance due to data noises and finite differenceapproximation errors. We provide a set of sufficient conditions which guaranteethat, from a single trajectory data denoised by a Local-Polynomial filter, thesupport of $\mathbf{c}(\lambda)$ asymptotically converges to the truesigned-support associated with the underlying PDE for sufficiently many dataand a certain range of $\lambda$. We also show various numerical experiments tovalidate our theory.

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