Online Weak-form Sparse Identification of Partial Differential Equations

This paper presents an online algorithm for identification of partial differential equations (PDEs) based on the weak-form sparse identification of nonlinear dynamics algorithm (WSINDy). The algorithm is online in the sense that if performs the identification task by processing solution snapshots that arrive sequentially. The core of the method combines a weak-form discretization of candidate PDEs with an online proximal gradient descent approach to the sparse regression problem. In particular, we do not regularize the ℓ0-pseudo-norm, instead finding that directly applying its proximal operator (which corresponds to a hard thresholding) leads to efficient online system identification from noisy data. We demonstrate the success of the method on the Kuramoto-Sivashinsky equation, the nonlinear wave equation with time-varying wavespeed, and the linear wave equation, in one, two, and three spatial dimensions, respectively. In particular, our examples show that the method is capable of identifying and tracking systems with coefficients that vary abruptly in time, and offers a streaming alternative to problems in higher dimensions.

本文提出了一种基于非线性动力学弱形式稀疏识别算法（WSINDy）的偏微分方程（PDEs）在线识别算法。该算法是在线的，即它通过依次处理到达的解快照来执行识别任务。该方法的核心是将候选偏微分方程的弱形式离散化与针对稀疏回归问题的在线近端梯度下降方法相结合。特别地，我们不对\(\ell_0\) -伪范数进行正则化，而是发现直接应用其近端算子（对应于硬阈值处理）能够从含噪数据中进行高效的在线系统识别。我们分别在一维、二维和三维空间中的Kuramoto - Sivashinsky方程、具有时变波速的非线性波动方程以及线性波动方程上证明了该方法的成功。特别地，我们的示例表明，该方法能够识别和跟踪系数随时间突然变化的系统，并为高维问题提供了一种流处理替代方案。