Weak SINDy for partial differential equations
Weak SINDy for partial differential equations
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DOI：
10.1016/j.jcp.2021.110525
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发表时间：
2021-07-14
影响因子：
4.1
通讯作者：
Bortz, David M.
中科院分区：
文献类型：
--
作者：
Messenger, Daniel A.;Bortz, David M.
非线性动力学稀疏识别（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力系统[6，39]。最近，几个小组独立地发现，弱公式提供了数量级更好的鲁棒性噪声。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使得能够从无噪声数据（即低于模拟方案的容差）有效地机器精度恢复模型系数，以及在大噪声状态下稳健地识别PDE（在许多众所周知的情况下，信噪比接近1）。这是通过离散化的卷积弱形式的PDE和利用分离性的测试功能，有效的模型识别使用快速傅立叶变换。所得到的偏微分方程的WSINDy算法具有O（ND+1 log（N））的最坏情况的计算复杂度的数据集与N个点在每个D + 1维。此外，我们的基于傅立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，我们利用测试函数的先验选择算法。最后，我们介绍了一个学习算法的阈值序列阈值最小二乘（STLS），使模型识别从大型图书馆，我们利用规模不变性在连续水平，以确定偏微分方程从规模不佳的数据集。我们证明了WSINDy的鲁棒性，速度和准确性在几个具有挑战性的PDE。代码在GitHub上公开，网址为https://github.com/MathBioCU/WSINDy_PDE。(C)2021爱思唯尔公司All rights reserved.
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 O(ND+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 sequentialthresholding 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. (C) 2021 Elsevier Inc. All rights reserved.