Asymptotic Theory of ℓ1-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 ℓ1 regularized Pseudo-Least Squares model~(ℓ1-PsLS). In any associative ℝ-algebra generated by finitely many differentiation operators that contain the unknown PDE operator, applying ℓ1-PsLS to a given data set yields a family of candidate models with coefficients c(λ) parameterized by the regularization weight λ≥ 0. The trace of c(λ)λ≥ 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 c(λ) asymptotically converges to the true signed-support associated with the underlying PDE for sufficiently many data and a certain range of λ. We also show various numerical experiments to validate our theory.