Weak SINDy For Partial Differential Equations

TL;DR

This paper extends the Weak SINDy framework to PDEs, enabling robust and efficient discovery of PDE models from noisy data using weak formulations, Fourier transforms, and adaptive algorithms.

Contribution

The paper introduces a PDE-specific Weak SINDy algorithm that improves noise robustness, computational efficiency, and model selection through novel Fourier-based methods and adaptive thresholding.

Findings

Effective noise robustness in PDE model discovery

Fast Fourier Transform accelerates the identification process

Robust performance demonstrated on challenging PDEs

Abstract

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.