  Classical model reduction techniques project the governing equations onto
linear subspaces of the high-dimensional state-space. For problems with slowly
decaying Kolmogorov-n-widths such as certain transport-dominated problems,
however, classical linear-subspace reduced-order models (ROMs) of low dimension
might yield inaccurate results. Thus, the concept of classical linear-subspace
ROMs has to be extended to more general concepts, like Model Order Reduction
(MOR) on manifolds. Moreover, as we are dealing with Hamiltonian systems, it is
crucial that the underlying symplectic structure is preserved in the reduced
model, as otherwise it could become unphysical in the sense that the energy is
not conserved or stability properties are lost. To the best of our knowledge,
existing literature addresses either MOR on manifolds or symplectic model
reduction for Hamiltonian systems, but not their combination. In this work, we
bridge the two aforementioned approaches by providing a novel projection
technique called symplectic manifold Galerkin (SMG), which projects the
Hamiltonian system onto a nonlinear symplectic trial manifold such that the
reduced model is again a Hamiltonian system. We derive analytical results such
as stability, energy-preservation and a rigorous a-posteriori error bound.
Moreover, we construct a weakly symplectic deep convolutional autoencoder as a
computationally practical approach to approximate a nonlinear symplectic trial
manifold. Finally, we numerically demonstrate the ability of the method to
outperform (non-)structure-preserving linear-subspace ROMs and
non-structure-preserving MOR on manifold techniques.
