Parametric high-fidelity simulations are of interest for a wide range of applications. But the restriction of computational resources renders such models to be inapplicable in a real-time context or in multi-query scenarios. Model order reduction (MOR) is used to tackle this issue. Recently, MOR is extended to preserve specific structures of the model throughout the reduction, e.g. structure-preserving MOR for Hamiltonian systems. This is referred to as symplectic MOR. It is based on the classical projection-based MOR and uses a symplectic reduced order basis (ROB). Such a ROB can be derived in a data-driven manner with the Proper Symplectic Decomposition (PSD) in the form of a minimization problem. Due to the strong nonlinearity of the minimization problem, it is unclear how to efficiently find a global optimum. In our paper, we show that current solution procedures almost exclusively yield suboptimal solutions by restricting to orthonormal ROBs. As new methodological contribution, we propose a new method which eliminates this restriction by generating non-orthonormal ROBs. In the numerical experiments, we examine the different techniques for a classical linear elasticity problem and observe that the non-orthonormal technique proposed in this paper shows superior results with respect to the error introduced by the reduction.For a data-driven generation of the ROB, the conventional methods e.g. the Proper Orthogonal Decomposition (POD) [4] are not suited since they do not necessarily compute a symplectic ROB. To this end, the referenced works introduce the Proper Symplectic Decomposition (PSD) which is a data-driven basis generation technique for symplectic ROBs. Due to the high nonlineariy of the optimization problem, an efficient solution strategy is yet unknown for the PSD. The existing PSD methods (Cotangent Lift, Complex SVD, a nonlinear programming approach [19] and a greedy procedure introduced in [15]) each restrict to a specific subset of symplectic ROBs from which they select optimal solutions which might be globally suboptimal.The present paper classifies the existing symplectic basis generation techniques in two classes of methods which either generate orthonormal or non-orthonormal bases. To this end, we show that the existing basis generation techniques for symplectic bases almost exclusively restrict to orthonormal bases. Furthermore, we prove that Complex SVD is the optimal solution of the PSD on the set of orthonormal, symplectic bases. During the proof, an alternative formulation of the Complex SVD for symplectic matrices is introduced. To leave the class of orthonormal, symplectic bases, we propose a new basis generation technique, namely the PSD SVD-like decomposition. It is based on an SVD-like decomposition of arbitrary matrices B ∈ R n×2m introduced in [21].This paper is organized in the following way: Section 2 is devoted to the structure-preserving MOR for autonomous and non-autonomous, parametric Hamiltonian systems and thus, introduces symplectic geometry, Hamiltonian systems and symple...
Classical model reduction techniques project the governing equations onto linear subspaces of the high-dimensional state-space. For problems with slowly decaying Kolmogorov-nwidths such as certain transport-dominated problems, however, classical linear-subspace reducedorder 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.
Neural networks can be used as surrogates for PDE models. They can be made physics-aware by penalizing underlying equations or the conservation of physical properties in the loss function during training. Current approaches allow to additionally respect data from numerical simulations or experiments in the training process. However, this data is frequently expensive to obtain and thus only scarcely available for complex models. In this work, we investigate how physics-aware models can be enriched with computationally cheaper, but inexact, data from other surrogate models like Reduced-Order Models (ROMs). In order to avoid trusting too-low-fidelity surrogate solutions, we develop an approach that is sensitive to the error in inexact data. As a proof of concept, we consider the one-dimensional wave equation and show that the training accuracy is increased by two orders of magnitude when inexact data from ROMs is incorporated.
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