Model reduction of port-Hamiltonian systems via neural networks

Abstract: In this paper, we consider structure-preserving model reduction of port-Hamiltonian (pH) systems which extend classical Hamiltonian systems with dissipation and an input-output port. These pH systems are often used in multi-physics systems, as the interconnection of one or more \pH systems results again in a pH system. If particularly the system matrices associated with the interconnection and/or dissipation of a pH system are state-dependent, then the evaluation of standard reduced-order models (ROMs) may depend on the dimension of the original full-order model, resulting in high computational costs. To circumvent these high costs, we propose to use structure-preserving neural networks. In particular, we perform two steps: (1) we use the generalized manifold Galerkin projection to project the pH system onto the reduced space; then (2) we train a neural network to learn the map from the reduced-order state to the reduced-order interconnection and dissipation system matrices. To ensure that the resulting ROM is again a pH system, the architecture of the neural network is chosen such that the skew-symmetry and positive semi-definiteness of the reduced-order systems matrices are maintained. In a numerical example, we consider a nonlinear mass-spring-damper system with state-dependent system matrices. The numerical results show that the proposed method achieves a significant computational speed-up compared to the original \ROM with comparable accuracy.