Extracting structured dynamical systems using sparse optimization with very few samples
Extracting structured dynamical systems using sparse optimization with very few samples
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DOI：
10.1137/18m1194730
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发表时间：
2018-05
期刊：
影响因子：
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通讯作者：
Hayden Schaeffer;Giang Tran;Rachel A. Ward;Linan Zhang
中科院分区：
文献类型：
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作者：
Hayden Schaeffer;Giang Tran;Rachel A. Ward;Linan Zhang
学习控制方程可以更深入地理解数据的结构和动态。我们提出了一种随机采样方法，用于从欠采样和可能有噪声的状态空间测量中学习结构化动力系统。学习问题的形式是一个稀疏的最小二乘拟合在一个大的候选函数集。基于部分相依随机变量的类Bernstein不等式，给出了稀疏系数的恢复率和相应问题的候选函数的确定的理论保证.计算结果显示在由Lorenz 96方程、粘性Burgers方程和双组分反应扩散方程（由于模型中的参数敏感性，这是具有挑战性的）生成的数据集上。该公式具有几个优点，包括易于使用，理论上的成功保证，以及相对于环境维度和候选函数的数量的计算效率。
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 under-sampled 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 (which is challenging due to parameter sensitives in the model). This formulation has several advantages including ease of use, theoretical guarantees of success, and computational efficiency with respect to ambient dimension and number of candidate functions.