  Sparse Identification of Nonlinear Dynamics (SINDy) is a method of system
discovery that has been shown to successfully recover governing dynamical
systems from data (Brunton et al., PNAS, '16; Rudy et al., Sci. Adv. '17).
Recently, several groups have independently discovered that the weak
formulation provides orders of magnitude better robustness to noise. Here we
extend our Weak SINDy (WSINDy) framework introduced in (arXiv:2005.04339) to
the setting of partial differential equations (PDEs). The elimination of
pointwise derivative approximations via the weak form enables effective
machine-precision recovery of model coefficients from noise-free data (i.e.
below the tolerance of the simulation scheme) as well as robust identification
of PDEs in the large noise regime (with signal-to-noise ratio approaching one
in many well-known cases). This is accomplished by discretizing a convolutional
weak form of the PDE and exploiting separability of test functions for
efficient model identification using the Fast Fourier Transform. The resulting
WSINDy algorithm for PDEs has a worst-case computational complexity of
$\mathcal{O}(N^{D+1}\log(N))$ for datasets with $N$ points in each of $D+1$
dimensions (i.e. $\mathcal{O}(\log(N))$ operations per datapoint). Furthermore,
our Fourier-based implementation reveals a connection between robustness to
noise and the spectra of test functions, which we utilize in an \textit{a
priori} selection algorithm for test functions. Finally, we introduce a
learning algorithm for the threshold in sequential-thresholding least-squares
(STLS) that enables model identification from large libraries, and we utilize
scale-invariance at the continuum level to identify PDEs from poorly-scaled
datasets. We demonstrate WSINDy's robustness, speed and accuracy on several
challenging PDEs.
