  Computing reduced-order models using non-intrusive methods is particularly
attractive for systems that are simulated using black-box solvers. However,
obtaining accurate data-driven models can be challenging, especially if the
underlying systems exhibit large-amplitude transient growth. Although these
systems may evolve near a low-dimensional subspace that can be easily
identified using standard techniques such as Proper Orthogonal Decomposition
(POD), computing accurate models often requires projecting the state onto this
subspace via a non-orthogonal projection. While appropriate oblique projection
operators can be computed using intrusive techniques that leverage the form of
the underlying governing equations, purely data-driven methods currently tend
to achieve dimensionality reduction via orthogonal projections, and this can
lead to models with poor predictive accuracy. In this paper, we address this
issue by introducing a non-intrusive framework designed to simultaneously
identify oblique projection operators and reduced-order dynamics. In
particular, given training trajectories and assuming reduced-order dynamics of
polynomial form, we fit a reduced-order model by solving an optimization
problem over the product manifold of a Grassmann manifold, a Stiefel manifold,
and several linear spaces (as many as the tensors that define the low-order
dynamics). Furthermore, we show that the gradient of the cost function with
respect to the optimization parameters can be conveniently written in
closed-form, so that there is no need for automatic differentiation. We compare
our formulation with state-of-the-art methods on three examples: a
three-dimensional system of ordinary differential equations, the complex
Ginzburg-Landau (CGL) equation, and a two-dimensional lid-driven cavity flow at
Reynolds number Re = 8300.
