  We demonstrate the synthesis of sparse sampling and machine learning to
characterize and model complex, nonlinear dynamical systems over a range of
bifurcation parameters. First, we construct modal libraries using the classical
proper orthogonal decomposition to uncover dominant low-rank coherent
structures. Here, nonlinear libraries are also constructed in order to take
advantage of the discrete empirical interpolation method and projection that
allows for the approximation of nonlinear terms in a low-dimensional way. The
selected sampling points are shown to be nearly optimal sensing locations for
characterizing the underlying dynamics, stability, and bifurcations of complex
systems. The use of empirical interpolation points and sparse representation
facilitate a family of local reduced-order models for each physical regime,
rather than a higher-order global model, which has the benefit of physical
interpretability of energy transfer between coherent structures. In particular,
the discrete interpolation points and nonlinear modal libraries are used for
sparse representation to classify the dynamic bifurcation regime in the complex
Ginzburg-Landau equation. It is shown that nonlinear point measurements are
more effective than linear measurements when sensor noise is present.
