Data-driven reduced-order models using quadratic manifolds

Data-driven methods such as dynamic mode decomposition (DMD) provide linear reduced-order models (ROMs) to approximate nonlinear systems. Typically the DMD-based ROMs perform well on a training data set; however, the ROMs' predictive accuracy is relatively poor, especially for complex nonlinear systems. In this talk, we propose a data-driven quadratic closure term for the DMD-based ROM, such that the resulting ROM has a linear-quadratic structure. Using the proposed closure term leads to a ROM whose trajectories evolve on a quadratic manifold; this is an improvement over the DMD-based ROM whose reduced-order trajectories evolve in a linear subspace. The quadratic manifold ROM display improved predictions and effectively approximates shocks and transient behavior. We demonstrate the enhancement of prediction capabilities on problems of traveling waves, the cylinder flow, and the lid-driven cavity. We also use the example of a traveling detonation wave in a shock tube to illustrate the ability of the proposed ROM to approximate shocks.