  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.
