In the beginning of this talk, we introduce the concept of model order reduction and provide a short overview of the field. Then, we consider so-called classical linear-subspace reduced order models (ROMs) and explain the necessary steps to arrive at such a model. Moreover, as we are dealing with Hamiltonian systems, it is crucial that we ensure that the underlying physical (here: symplectic) structure is preserved in the reduced model.
In the second part of the talk, we consider problems, where classical linear-subspace ROMs of low dimension might yield inaccurate results. This could be the case for certain transport-dominated problems. Thus, the reduced space needs to be extended to more general nonlinear manifolds. To this end, we provide a novel projection technique called symplectic manifold Galerkin. We derive analytical results such as stability, energy-preservation and a rigorous a-posteriori error bound. Furthermore, we provide methods to computationally approximate the nonlinear symplectic trial manifold and numerically demonstrate the ability of the method to outperform structure-preserving linear subspace ROMs.