Structure-Preserving Galerkin POD-DEIM Reduced-Order Modeling of Hamiltonian Systems

2021/02/28 by Zhu Wang, Wang, Zhu · 2 citations
Engineering · Mathematics · Physics and Astronomy · #37M25 #65M99 #65P10 #93A15 #FOS: Mathematics #Model Reduction and Neural Networks #Numerical Analysis (math.NA) #Numerical methods for differential equations #Real-time simulation and control systems

paper · pdf · doi:10.48550/arxiv.2103.00388

openalex publication_date 2021/02/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A structure preserving proper orthogonal decomposition reduce-order modeling approach has been developed in [Gong et al. 2017] for the Hamiltonian system, which uses the traditional framework of Galerkin projection-based model reduction but modifies the reduced order model so that the appropriate Hamiltonian structure is preserved. However, its computational complexity for online simulations is still high if the Hamiltonian involves non-polynomial nonlinearities. In this paper, we apply the discrete empirical interpolation method to improve the online efficiency of the structure-preserving reduced order simulations. Since the reduced basis truncation can degrade the Hamiltonian approximation, we propose to use the basis obtained from shifted snapshots. A nonlinear wave equation is used as a test bed and the numerical results illustrate the efficacy of the proposed method.

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• Structure-preserving Lift & Learn: Scientific machine learning for nonlinear conservative partial differential equations
• Hamiltonian reduction using a convolutional auto-encoder coupled to an Hamiltonian neural network

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