  This paper is on the construction of structure-preserving, online-efficient
reduced models for the barotropic Euler equations with a friction term on
networks. The nonlinear flow problem finds broad application in the context of
gas distribution networks. We propose a snapshot-based reduction approach that
consists of a mixed variational Galerkin approximation combined with
quadrature-type complexity reduction. Its main feature is that certain
compatibility conditions are assured during the training phase, which make our
approach structure-preserving. The resulting reduced models are locally mass
conservative and inherit an energy-bound and port-Hamiltonian structure. We
also derive a well-posedness result for them. In the training phase, the
compatibility conditions pose challenges, we face constrained data
approximation problems as opposed to the unconstrained training problems in the
conventional reduction methods. The training of our model order reduction
consists of a principal component analysis under a compatibility constraint
and, notably, yields reduced models that fulfill an optimality condition for
the snapshot data. The training of our quadrature-type complexity reduction
involves a semi-definite program with combinatorial aspects, which we approach
by a greedy procedure.
