Messenger Daniel A, Bortz David M
Department of Applied Mathematics, University of Colorado Boulder, 11 Engineering Dr., Boulder, CO 80309, USA.
J Comput Phys. 2021 Oct 15;443. doi: 10.1016/j.jcp.2021.110525. Epub 2021 Jun 23.
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 for datasets with points in each of + 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 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，已被证明能成功地从数据中恢复主导动力学系统[6, 39]。最近，几个研究团队独立发现，弱形式对噪声的鲁棒性提高了几个数量级。在此，我们将在[28]中引入的弱SINDy（WSINDy）框架扩展到偏微分方程（PDE）的情形。通过弱形式消除逐点导数近似，能够从无噪声数据（即低于模拟方案的容差）中有效地以机器精度恢复模型系数，并且在大噪声 regime（在许多著名情况下信噪比接近1）中对PDE进行鲁棒识别。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高效模型识别来实现的。对于在每个d + 1维度中有N个点的数据集，所得的PDE的WSINDy算法具有最坏情况的计算复杂度为 。此外，我们基于傅里叶的实现揭示了对噪声的鲁棒性与测试函数谱之间的联系，我们在测试函数的 选择算法中利用了这一点。最后，我们为顺序阈值最小二乘法（STLS）中的阈值引入了一种学习算法，该算法能够从大型库中进行模型识别，并且我们利用连续水平的尺度不变性从尺度不佳的数据集中识别PDE。我们在几个具有挑战性的PDE上展示了WSINDy的鲁棒性、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 。
