AI 中文总结
本文针对状态依赖系统矩阵的端口哈密顿系统，采用广义流形伽辽金投影结合保结构神经网络，提出保结构模型降阶方法，实现了精度相当下的显著计算加速。
AI 中文摘要
本文研究端口哈密顿（pH）系统的保结构模型降阶，这类系统在经典哈密顿系统基础上扩展了耗散项与输入输出端口，常用于多物理场系统，且多个pH系统互连后仍为pH系统。若pH系统中与互连和/或耗散相关的系统矩阵依赖于状态，标准降阶模型（ROM）的计算会依赖原始全阶模型的维度，导致计算成本过高。为规避该问题，本文提出使用保结构神经网络，具体分为两步：（1）采用广义流形伽辽金投影将pH系统投影到降阶空间；（2）训练神经网络学习从降阶状态到降阶互连与耗散系统矩阵的映射。为确保所得ROM仍为pH系统，神经网络的架构设计需维持降阶系统矩阵的斜对称性与半正定性。数值算例中，本文考虑了系统矩阵依赖于状态的非线性质量-弹簧-阻尼系统，结果表明，与原始ROM相比，所提方法在精度相当的情况下实现了显著的计算加速。
英文摘要
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.
