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.
翻译:经典模型削减技术将治理方程式投射到高维状态空间的线性子空间上。对于缓慢衰落的科尔莫戈洛夫-恩维特(Kolmogorov-n-width)的问题,例如某些运输占主导地位的问题,然而,对于典型的线性子空间降序模型(ROMs),低维的经典线性亚空间降序模型(ROMs)可能会产生不准确的结果。因此,经典线性亚空间ROM的概念必须扩展至更一般的概念,如模型降序(MOR)在多层上。此外,在我们处理汉密尔密尔顿系统时,根本的静态结构(SMGMG)将汉密尔顿系统保留在非线性模式上,否则它可能变得不物理,因为能源不会被保存或稳定性特性丧失。根据我们的知识,现有的文献要么在模型上,光线性子-线性子-子系统会将精度的精度降精度的精度精确度结构,我们将精度的精度的精度机精度的精度的精度的精度的精度变的精度模型将精度的精度机的精度构造构造构造构造 进的精细的精细的精细的精细的精细的精细的精细的精细的精细的精细的精细的精细的精细的精细的精细的精细的精细的研细的精细的精细的精细的精细的精细的精细的精细的精细的精细的精细的研细的精细的精细的精细的精细的精细的精细的精细的精细的精细的精细的精细的精细的精细的精细的精细的精细的精细的精细的精细的精细的精细的精细的精细的精细的精细的精细的精细的精细的精细的精细的精细的精细的精细的精细的精细的精细的精细的精细的精细的精细的精细的精细的精细的精细的精细的精细的精细的精细的精细的精细的精细