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

TL;DR

This paper establishes the asymptotic support recovery of unknown PDEs from a single noisy trajectory using $$-regularized pseudo-least squares, under certain conditions, with validation through numerical experiments.

Contribution

It provides the first rigorous asymptotic theory for PDE identification from noisy data using $$ regularization, including support recovery guarantees.

Findings

Support of coefficients converges to true support with enough data.

Denoising with local polynomial filters is effective.

Numerical experiments validate theoretical results.

Abstract

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 $\lambda\geq 0$. The trace of $\{\mathbf{c}(\lambda)\}_{\lambda\geq 0}$ 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.