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In this work we study the asymptotic consistency of the weak-form sparseidentification of nonlinear dynamics algorithm (WSINDy) in the identificationof differential equations from noisy samples of solutions. We prove that theWSINDy estimator is unconditionally asymptotically consistent for a wide classof models which includes the Navier-Stokes equations and theKuramoto-Sivashinsky equation. We thus provide a mathematically rigorousexplanation for the observed robustness to noise of weak-form equationlearning. Conversely, we also show that in general the WSINDy estimator is onlyconditionally asymptotically consistent, yielding discovery of spurious termswith probability one if the noise level is above some critical threshold andthe nonlinearities exhibit sufficiently fast growth. We derive explicit boundson the critical noise threshold in the case of Gaussian white noise and providean explicit characterization of these spurious terms in the case oftrigonometric and/or polynomial model nonlinearities. However, a silver liningto this negative result is that if the data is suitably denoised (a simplemoving average filter is sufficient), then we recover unconditional asymptoticconsistency on the class of models with locally-Lipschitz nonlinearities.Altogether, our results reveal several important aspects of weak-form equationlearning which may be used to improve future algorithms. We demonstrate ourresults numerically using the Lorenz system, the cubic oscillator, a viscousBurgers growth model, and a Kuramoto-Sivashinsky-type higher-order PDE.

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