  Extracting governing equations from dynamic data is an essential task in
model selection and parameter estimation. The form of the governing equation is
rarely known a priori; however, based on the sparsity-of-effect principle one
may assume that the number of candidate functions needed to represent the
dynamics is very small. In this work, we leverage the sparse structure of the
governing equations along with recent results from random sampling theory to
develop methods for selecting dynamical systems from under-sampled data. In
particular, we detail three sampling strategies that lead to the exact recovery
of first-order dynamical systems when we are given fewer samples than unknowns.
The first method makes no assumptions on the behavior of the data, and requires
a certain number of random initial samples. The second method utilizes the
structure of the governing equation to limit the number of random
initializations needed. The third method leverages chaotic behavior in the data
to construct a nearly deterministic sampling strategy. Using results from
compressive sensing, we show that the strategies lead to exact recovery, which
is stable to the sparse structure of the governing equations and robust to
noise in the estimation of the velocity. Computational results validate each of
the sampling strategies and highlight potential applications.
