  Characterizing and controlling nonlinear, multi-scale phenomena play
important roles in science and engineering. Cluster-based reduced-order
modeling (CROM) was introduced to exploit the underlying low-dimensional
dynamics of complex systems. CROM builds a data-driven discretization of the
Perron-Frobenius operator, resulting in a probabilistic model for ensembles of
trajectories. A key advantage of CROM is that it embeds nonlinear dynamics in a
linear framework, and uncertainty can be managed with data assimilation. CROM
is typically computed on high-dimensional data, however, access to and
computations on this full-state data limit the online implementation of CROM
for prediction and control. Here, we address this key challenge by identifying
a small subset of critical measurements to learn an efficient CROM, referred to
as sparsity-enabled CROM. In particular, we leverage compressive measurements
to faithfully embed the cluster geometry and preserve the probabilistic
dynamics. Further, we show how to identify fewer optimized sensor locations
tailored to a specific problem that outperform random measurements. Both of
these sparsity-enabled sensing strategies significantly reduce the burden of
data acquisition and processing for low-latency in-time estimation and control.
We illustrate this unsupervised learning approach on three different
high-dimensional nonlinear dynamical systems from fluids with increasing
complexity, with one application in flow control. Sparsity-enabled CROM is a
critical facilitator for real-time implementation on high-dimensional systems
where full-state information may be inaccessible.
