  Data-driven reduced-order models often fail to make accurate forecasts of
high-dimensional nonlinear dynamical systems that are sensitive along
coordinates with low-variance because such coordinates are often truncated,
e.g., by proper orthogonal decomposition, kernel principal component analysis,
and autoencoders. Such systems are encountered frequently in shear-dominated
fluid flows where non-normality plays a significant role in the growth of
disturbances. In order to address these issues, we employ ideas from active
subspaces to find low-dimensional systems of coordinates for model reduction
that balance adjoint-based information about the system's sensitivity with the
variance of states along trajectories. The resulting method, which we refer to
as covariance balancing reduction using adjoint snapshots (CoBRAS), is
analogous to balanced truncation with state and adjoint-based gradient
covariance matrices replacing the system Gramians and obeying the same key
transformation laws. Here, the extracted coordinates are associated with an
oblique projection that can be used to construct Petrov-Galerkin reduced-order
models. We provide an efficient snapshot-based computational method analogous
to balanced proper orthogonal decomposition. This also leads to the observation
that the reduced coordinates can be computed relying on inner products of state
and gradient samples alone, allowing us to find rich nonlinear coordinates by
replacing the inner product with a kernel function. In these coordinates,
reduced-order models can be learned using regression. We demonstrate these
techniques and compare to a variety of other methods on a simple, yet
challenging three-dimensional system and a nonlinear axisymmetric jet flow
simulation with $10^5$ state variables.
