  We propose a general dynamic reduced-order modeling framework for typical
experimental data: time-resolved sensor data and optional non-time-resolved PIV
snapshots. This framework contains four steps. First, the sensor signals are
lifted to a dynamic feature space. Second, we identify a sparse
human-interpretable nonlinear dynamical system for the feature state based on
the sparse identification of nonlinear dynamics (SINDy). Third, if PIV
snapshots are available, a local linear mapping from the feature state to
velocity fields is shown to be orders of magnitudes more accurate than optimal
modal expansions of the same order. Fourth, a generalized feature-based modal
decomposition identifies coherent structures that are most dynamically
correlated with the linear and nonlinear interaction terms in the sparse model,
adding interpretability. Steps 1 and 2 define a black-box model. Optional steps
3 and 4 lift the black-box dynamics to a 'gray-box' model of the coherent
structures, if non-time-resolved full-state data is available. This gray-box
modeling strategy is successfully applied to the transient and post-transient
laminar cylinder wake, and compares favorably with a POD model. We foresee
numerous applications of this highly flexible modeling strategy, including
estimation, prediction and control. Moreover, the feature space may be based on
intrinsic coordinates, which are unaffected by a key challenge of modal
expansion: the slow change of low-dimensional coherent structures with changing
geometry and varying parameters.
