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：本文讨论了基于全阶状态通过一组依赖于降阶模型状态的基函数线性组合近似的端口-哈密顿系统保结构模型降阶方法。近年来，这类非线性近似基因在输运主导系统（传统线性模型降阶技术难以处理的系统）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，通常可以获得端口-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，通过两个输运主导的数值算例验证了该方法，其中基函数基于全阶状态的快照数据确定。