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