# Asymptotic Theory of ℓ1-Regularized PDE Identification from a Single Noisy Trajectory

arXiv: https://arxiv.org/abs/2103.07045 · HTML: https://mlsanity.com/paper/2103.07045/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 2 of its 64 references that are in the map.
- First posted on arXiv 12 Mar 2021.

## Builds on (most cited first, 2 of 2)

- [Extracting Sparse High-Dimensional Dynamics from Limited Data](https://mlsanity.com/paper/1707.08528/) (2017)
- [Sparse Identification of Nonlinear Dynamical Systems via Reweighted ℓ1-regularized Least Squares](https://mlsanity.com/paper/2005.13232/) (2020)

## Abstract (arXiv)

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.