Symplectic Model Reduction of Hamiltonian Systems on Nonlinear Manifolds

• DOI: 10.48550/arxiv.2112.10815
• Publication Date: 2021-01-01
• Authors: Patrick Buchfink; Silke Glas; Bernard Haasdonk
• Keywords: Hamiltonian (control theory); Manifold (fluid mechanics); Center manifold

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