Optimizing Oblique Projections for Nonlinear Systems using Trajectories

Reduced-order modeling techniques, including balanced truncation and $\mathcal{H}_2$-optimal model reduction, exploit the structure of linear dynamical systems to produce models that accurately capture dynamics. For nonlinear operating far away from equilibria, on other hand, current approaches seek low-dimensional representations state often neglect low-energy features have high significance. instance, are known play an important role in fluid dynamics, where they can be a driving mechanism for shear-layer instabilities. Neglecting these leads with poor predictive accuracy despite being able encode decode states. In order improve accuracy, we propose optimize reduced-order fit collection coarsely sampled trajectories original system. particular, over product two Grassmann manifolds defining Petrov--Galerkin projections full-order governing equations. We compare our approach existing methods proper orthogonal decomposition, truncation-based projection, quadratic-bilinear truncation, iterative rational Krylov algorithm. Our demonstrates significantly improved both toy incompressible (nonlinear) axisymmetric jet flow $10^5$