Sparse Identification of Nonlinear Dynamics (SINDy) is a method of system discovery that has been shown to successfully recover governing dynamical systems from data [6], [39]. 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 [28] 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. Furthermore, our Fourier-based implementation reveals a connection between robustness to noise and the spectra of test functions, which we utilize in an 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. Code is publicly available on GitHub at https://github.com/MathBioCU/WSINDy_PDE.

非线性动力学的稀疏识别 (SINDy) 是一种系统发现方法，已被证明可以成功地从数据中恢复控制动态系统 [6]、[39]。最近，几个小组独立发现，弱公式对噪声的鲁棒性要好几个数量级。在这里，我们将 [28] 中介绍的弱 SINDy (WSINDy) 框架扩展到偏微分方程 (PDE) 的设置。通过弱形式消除逐点导数近似可以有效地从无噪声数据中恢复模型系数（即低于模拟方案的容差），以及在大噪声状态下稳健地识别偏微分方程（具有信号-在许多众所周知的情况下，信噪比接近 1）。这是通过离散化 PDE 的卷积弱形式并利用测试函数的可分离性使用快速傅里叶变换进行有效模型识别来实现的。生成的 PDE 的 WSINDy 算法的最坏情况计算复杂度为$\mathcal{○}({ñ}^{D+1}\mathrm{日志}(ñ))$对于每个数据集中有N个点的数据集$D+1$方面。此外，我们基于傅里叶的实现揭示了对噪声的鲁棒性和测试函数的频谱之间的联系，我们在测试函数的先验选择算法中使用了它。最后，我们在顺序阈值最小二乘法 (STLS) 中引入了一种阈值学习算法，该算法能够从大型库中识别模型，并且我们利用连续体级别的尺度不变性来从尺度较差的数据集中识别 PDE。我们在几个具有挑战性的 PDE 上展示了 WSINDy 的稳健性、速度和准确性。代码可在 GitHub 上公开获取，网址为 https://github.com/MathBioCU/WSINDy_PDE。