Asymptotic consistency of the WSINDy algorithm in the limit of continuum data

TL;DR

This paper rigorously analyzes the asymptotic consistency of the WSINDy algorithm for identifying differential equations from noisy data, showing conditions for robustness and the impact of noise on spurious term discovery.

Contribution

It provides a mathematical proof of when WSINDy is unconditionally consistent and characterizes the effects of noise, including bounds and conditions for recovering true models.

Findings

WSINDy is unconditionally consistent for certain models like Navier-Stokes.

High noise levels can lead to spurious term discovery without denoising.

Simple denoising restores unconditional consistency for models with Lipschitz nonlinearities.

Abstract

In this work we study the asymptotic consistency of the weak-form sparse identification of nonlinear dynamics algorithm (WSINDy) in the identification of differential equations from noisy samples of solutions. We prove that the WSINDy estimator is unconditionally asymptotically consistent for a wide class of models which includes the Navier-Stokes equations and the Kuramoto-Sivashinsky equation. We thus provide a mathematically rigorous explanation for the observed robustness to noise of weak-form equation learning. Conversely, we also show that in general the WSINDy estimator is only conditionally asymptotically consistent, yielding discovery of spurious terms with probability one if the noise level is above some critical threshold and the nonlinearities exhibit sufficiently fast growth. We derive explicit bounds on the critical noise threshold in the case of Gaussian white noise and provide an explicit characterization of these spurious terms in the case of trigonometric and/or polynomial model nonlinearities. However, a silver lining to this negative result is that if the data is suitably denoised (a simple moving average filter is sufficient), then we recover unconditional asymptotic consistency on the class of models with locally-Lipschitz nonlinearities. Altogether, our results reveal several important aspects of weak-form equation learning which may be used to improve future algorithms. We demonstrate our results numerically using the Lorenz system, the cubic oscillator, a viscous Burgers growth model, and a Kuramoto-Sivashinsky-type higher-order PDE.