论文标题
偏微分方程的弱信也不足
Weak SINDy For Partial Differential Equations
论文作者
论文摘要
非线性动力学（Sindy）的稀疏识别是一种系统发现的方法，已证明可以从数据中成功恢复控制动态系统（Brunton等，PNAS，16； Rudy等，Sci。Adv。'17）。最近，几个小组独立地发现，弱公式为噪声提供了更好的稳健性。在这里，我们将（Arxiv：2005.04339）引入的弱信也扩展到偏微分方程（PDE）的设置。通过弱形式消除了点式衍生近似值，可以从无噪声数据（即在模拟方案的公差下）进行有效的机器前置恢复，以及在大噪声状态下对PDE的强大鉴定（在许多众所周知的情况下，在较大的噪声比率接近信号噪声比率）中对PDE进行了强有力的识别）。这是通过离散PDE的卷积弱形式来实现的，并利用快速的傅立叶变换来利用测试函数的可分离性，以进行有效的模型识别。 PDE的生成的WSINDY算法的最差计算复杂性为$ \ MATHCAL {O}（n^{d+1} \ log（n））$，用于$ d+1 $ dimens的$ n $点的数据集（即$ d+1 $ dimensions（即$ \ \ mathcal {o} o}（o}（o log log log（n））$（\ log log（n））。此外，我们的基于傅立叶的实现揭示了稳健性与噪声与测试函数的光谱之间的联系，我们在\ textIt {a a先验}选择算法中用于测试功能。最后，我们引入了一种学习算法，以实现顺序阈值最小二乘（STLS）的阈值，该算法可以从大型库中识别模型识别，并且我们利用连续级别的规模不变性来识别较差的数据集中的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。
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.
