Asymptotic Theory of ℓ_1-Regularized PDE Identification from a Single Noisy Trajectory
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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 ℓ_1 regularized Pseudo-Least Squares model (ℓ_1-PsLS). In any associative ℝ-algebra generated by finitely many differentiation operators that contain the unknown PDE operator, applying ℓ_1-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.
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