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：自动模型选择是科学与工程中的一项重要应用。本文针对欠采样且含噪声的时空数据，提出了一种识别结构化动力系统的学习方法。该学习通过大量候选函数上的稀疏最小二乘拟合实现，并采用交替方向乘子法求解非凸 $\ell_1-\ell_2$ 稀疏优化问题。利用伯恩斯坦型不等式与相干性条件，我们证明了：若候选函数集构成有界正交系统上的结构化随机采样矩阵，则恢复过程稳定且误差有界。该学习方法在粘性Burgers方程及两个反应扩散方程生成的合成数据上得到了验证。计算结果展示了理论保证的成功性，以及该方法在环境维度和候选函数数量方面的效率。