We discuss structure-preserving model order reduction for port-Hamiltonian systems based on an approximation of the full-order state by a linear combination of ansatz functions which depend themselves on the state of the reduced-order model. In recent years, such nonlinear approximation ansatzes have gained more and more attention especially due to their effectiveness in the context of model reduction for transport-dominated systems which are challenging for classical linear model reduction techniques. We demonstrate that port-Hamiltonian reduced-order models can often be obtained by a residual minimization approach where a special weighted norm is used for the residual. Moreover, we discuss sufficient conditions for the resulting reduced-order models to be stable. Finally, the methodology is illustrated by means of two transport-dominated numerical test cases, where the ansatz functions are determined based on snapshot data of the full-order state.

翻译：本文讨论基于全阶状态由降阶模型状态自身依赖的Ansatz函数线性组合近似的端口哈密顿系统保结构模型降阶。近年来，此类非线性近似Ansatz因其在输运主导系统（传统线性模型降阶技术难以处理）模型降阶中的有效性而受到越来越多的关注。我们证明，通过采用特殊加权范数的残差最小化方法，通常可获得端口哈密顿降阶模型。此外，我们讨论了所得降阶模型稳定的充分条件。最后，通过两个输运主导数值测试案例验证了该方法的有效性，其中Ansatz函数基于全阶状态的快照数据确定。