We propose robust methods to identify underlying Partial Differential Equation (PDE) from a given set of noisy time dependent data. We assume that the governing equation is a linear combination of a few linear and nonlinear differential terms in a prescribed dictionary. Noisy data make such identification particularly challenging. Our objective is to develop methods which are robust against a high level of noise, and to approximate the underlying noise-free dynamics well. We first introduce a Successively Denoised Differentiation (SDD) scheme to stabilize the amplified noise in numerical differentiation. SDD effectively denoises the given data and the corresponding derivatives. Secondly, we present two algorithms for PDE identification: Subspace pursuit Time evolution error (ST) and Subspace pursuit Cross-validation (SC). Our general strategy is to first find a candidate set using the Subspace Pursuit (SP) greedy algorithm, then choose the best one via time evolution or cross validation. ST uses multi-shooting numerical time evolution and selects the PDE which yields the least evolution error. SC evaluates the cross-validation error in the least squares fitting and picks the PDE that gives the smallest validation error. We present a unified notion of PDE identification error to compare the objectives of related approaches. We present various numerical experiments to validate our methods. Both methods are efficient and robust to noise.

翻译：本文提出从含噪时间依赖数据中识别底层偏微分方程的鲁棒方法。我们假设控制方程是预设字典中若干线性与非线性微分项的线性组合。噪声数据使得此类识别任务极具挑战性。本研究目标在于开发对高噪声水平具有鲁棒性的方法，并有效逼近无噪声动力学过程。首先引入逐次去噪微分（SDD）方案以稳定数值微分中放大的噪声。SDD可有效对原始数据及其对应导数进行去噪处理。其次提出两种偏微分方程识别算法：子空间追踪时间演化误差（ST）与子空间追踪交叉验证（SC）。整体策略为先利用子空间追踪（SP）贪婪算法获取候选集，再通过时间演化或交叉验证选取最优方程。ST采用多次打靶数值时间演化，选取产生最小演化误差的偏微分方程；SC通过最小二乘拟合评估交叉验证误差，选取验证误差最小的偏微分方程。我们提出统一的偏微分方程识别误差概念用于比较相关方法的目标。通过多种数值实验验证了所提方法的有效性。两种方法均具有高效性与噪声鲁棒性。