Automated model selection is an important application in science and engineering. In this work, we develop a learning approach for identifying structured dynamical systems from undersampled and noisy spatiotemporal data. The learning is performed by a sparse least-squares fitting over a large set of candidate functions via a nonconvex $\ell_1-\ell_2$ sparse optimization solved by the alternating direction method of multipliers. Using a Bernstein-like inequality with a coherence condition, we show that if the set of candidate functions forms a structured random sampling matrix of a bounded orthogonal system, the recovery is stable and the error is bounded. The learning approach is validated on synthetic data generated by the viscous Burgers' equation and two reaction-diffusion equations. The computational results demonstrate the theoretical guarantees of success and the efficiency with respect to the ambient dimension and the number of candidate functions.

翻译：自动化的模型选择是科学与工程领域中的一项重要应用。本文开发了一种学习方法，用于从欠采样和含噪声的时空数据中识别结构化动力系统。该方法通过稀疏最小二乘拟合来学习，其核心是在大量候选函数集合上利用交替方向乘子法求解非凸$\ell_1-\ell_2$稀疏优化问题。利用带有相干性条件的Bernstein型不等式，我们证明了当候选函数集合构成有界正交系统的结构化随机采样矩阵时，该恢复过程具有稳定性且误差有界。该学习方法在由粘性Burgers方程和两个反应扩散方程生成的合成数据上得到验证。计算结果证实了理论上的成功保证，以及该方法在环境维度和候选函数数量方面的效率。