Sparse reduced-order modelling: sensor-based dynamics to full-state estimation

We propose a general dynamic reduced-order modelling framework for typical experimental data: time-resolved sensor data and optional non-time-resolved particle image velocimetry (PIV) snapshots. This framework can be decomposed into four building blocks. First, the sensor signals are lifted to a dynamic feature space without false neighbours. 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 the velocity field is performed to reconstruct the full state of the system. 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 grey-box model in terms of the identified coherent structures, if non-time-resolved full-state data are available. This grey-box modelling strategy is successfully applied to the transient and post-transient laminar cylinder wake, and compares favourably with a proper orthogonal decomposition model. We foresee numerous applications of this highly flexible modelling 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.