Hamiltonian operator inference has been developed in [Sharma, H., Wang, Z., Kramer, B., Physica D: Nonlinear Phenomena, 431, p.133122, 2022] to learn structure-preserving reduced-order models (ROMs) for Hamiltonian systems. The method constructs a low-dimensional model using only data and knowledge of the functional form of the Hamiltonian. The resulting ROMs preserve the intrinsic structure of the system, ensuring that the mechanical and physical properties of the system are maintained. In this work, we extend this approach to port-Hamiltonian systems, which generalize Hamiltonian systems by including energy dissipation, external input, and output. Based on snapshots of the system's state and output, together with the information about the functional form of the Hamiltonian, reduced operators are inferred through optimization and are then used to construct data-driven ROMs. To further alleviate the complexity of evaluating nonlinear terms in the ROMs, a hyper-reduction method via discrete empirical interpolation is applied. Accordingly, we derive error estimates for the ROM approximations of the state and output. Finally, we demonstrate the structure preservation, as well as the accuracy of the proposed port-Hamiltonian operator inference framework, through numerical experiments on a linear mass-spring-damper problem and a nonlinear Toda lattice problem.

翻译：哈密顿算子推断方法已在[Sharma, H., Wang, Z., Kramer, B., Physica D: Nonlinear Phenomena, 431, p.133122, 2022]中提出，用于为哈密顿系统学习保持结构的降阶模型。该方法仅利用数据和哈密顿量函数形式的知识构建低维模型。所得降阶模型保持了系统的内在结构，确保系统的力学与物理特性得以保留。本工作将此方法推广至端口-哈密顿系统——该系统通过纳入能量耗散、外部输入与输出，推广了经典哈密顿系统。基于系统状态与输出的快照数据，结合哈密顿量函数形式信息，通过优化推断出降阶算子，进而构建数据驱动的降阶模型。为缓解降阶模型中非线性项求值的复杂度，采用基于离散经验插值的超降阶方法。据此，我们推导了状态与输出降阶逼近的误差估计。最后，通过线性质量-弹簧-阻尼器问题与非线性Toda晶格问题的数值实验，验证了所提端口-哈密顿算子推断框架的结构保持特性与计算精度。