  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.
