Multiscale Modeling &Simulation, Volume 18, Issue 4, Page 1435-1461, January 2020.
Learning governing equations allows for deeper understanding of the structure and dynamics of data. We present a random sampling method for learning structured dynamical systems from undersampled and possibly noisy state-space measurements. The learning problem takes the form of a sparse least-squares fitting over a large set of candidate functions. Based on a Bernstein-like inequality for partly dependent random variables, we provide theoretical guarantees on the recovery rate of the sparse coefficients and the identification of the candidate functions for the corresponding problem. Computational results are demonstrated on datasets generated by the Lorenz 96 equation, the viscous Burgers' equation, and the two-component reaction-diffusion equations. Our formulation includes theoretical guarantees of success and is shown to be efficient with respect to the ambient dimension and the number of candidate functions.

中文翻译：

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使用很少样本的稀疏优化来提取结构动力系统

2020年1月，《多尺度建模与仿真》，第18卷，第4期，第1435-1461页。
学习控制方程式可以更深入地了解数据的结构和动力学。我们提出了一种随机采样方法，用于从欠采样和可能带有噪声的状态空间测量中学习结构化动力系统。学习问题采取的是稀疏最小二乘拟合在大量候选函数上的形式。基于部分相关随机变量的类Bernstein不等式，我们为稀疏系数的恢复率和相应问题的候选函数的识别提供了理论保证。计算结果在Lorenz 96方程，粘性Burgers方程和两组分反应扩散方程生成的数据集上得到证明。