  Nearly all model-reduction techniques project the governing equations onto a
linear subspace of the original state space. Such subspaces are typically
computed using methods such as balanced truncation, rational interpolation, the
reduced-basis method, and (balanced) POD. Unfortunately, restricting the state
to evolve in a linear subspace imposes a fundamental limitation to the accuracy
of the resulting reduced-order model (ROM). In particular, linear-subspace ROMs
can be expected to produce low-dimensional models with high accuracy only if
the problem admits a fast decaying Kolmogorov $n$-width (e.g.,
diffusion-dominated problems). Unfortunately, many problems of interest exhibit
a slowly decaying Kolmogorov $n$-width (e.g., advection-dominated problems). To
address this, we propose a novel framework for projecting dynamical systems
onto nonlinear manifolds using minimum-residual formulations at the
time-continuous and time-discrete levels; the former leads to manifold Galerkin
projection, while the latter leads to manifold least-squares Petrov--Galerkin
(LSPG) projection. We perform analyses that provide insight into the
relationship between these proposed approaches and classical linear-subspace
reduced-order models; we also derive a posteriori discrete-time error bounds
for the proposed approaches. In addition, we propose a computationally
practical approach for computing the nonlinear manifold, which is based on
convolutional autoencoders from deep learning. Finally, we demonstrate the
ability of the method to significantly outperform even the optimal
linear-subspace ROM on benchmark advection-dominated problems, thereby
demonstrating the method's ability to overcome the intrinsic $n$-width
limitations of linear subspaces.
