Weak SINDy For Partial Differential Equations

We extend the WSINDy (Weak SINDy) method of sparse recovery introduced previously by the authors (arXiv:2005.04339) to the setting of partial differential equations (PDEs). As in the case of ODE discovery, the weak form replaces pointwise approximation of derivatives with local integrations against test functions and achieves effective machine-precision recovery of weights from noise-free data (i.e. below the tolerance of the simulation scheme) as well as natural robustness to noise without the use of noise filtering. The resulting WSINDy_PDE algorithm uses separable test functions implemented efficiently via convolutions for discovery of PDE models with computational complexity $O(NM)$ from data points with $M = N^{D+1}$ points, or $N$ points in each of $D+1$ dimensions. We demonstrate on several notoriously challenging PDEs the speed and accuracy with which WSINDy_PDE recovers the correct models from datasets with surprisingly large levels noise (often with levels of noise much greater than 10%).