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%).

中文翻译：

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偏微分方程的弱 SINDy

我们将作者 (arXiv:2005.04339) 之前介绍的 WSINDy（Weak SINDy）稀疏恢复方法扩展到偏微分方程 (PDE) 的设置。与 ODE 发现的情况一样，弱形式用针对测试函数的局部积分代替导数的逐点逼近，并从无噪声数据（即低于模拟方案的容差）和自然数据中实现有效的机器精度权重恢复在不使用噪声过滤的情况下对噪声具有鲁棒性。由此产生的 WSINDy_PDE 算法使用通过卷积有效实现的可分离测试函数，以从具有 $M = N^{D+1}$ 点的数据点中发现计算复杂度为 $O(NM)$ 的 PDE 模型，或每个点中的 $N$ 点$D+1$ 尺寸。