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

• Main Authors: He, Yuchen, Suh, Namjoon, Huo, Xiaoming, Kang, Sungha, Mei, Yajun
• Format: Journal Article
• Language: English
• Published: 11-03-2021
• Subjects: Computer Science - Numerical Analysis; Mathematics - Numerical Analysis; Statistics - Machine Learning
• DOI: 10.48550/arxiv.2103.07045

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.