  The Kolmogorov $n$-width is an established benchmark to judge the performance
of reduced basis and similar methods that produce linear reduced spaces.
Although immensely successful in the elliptic regime, this width, shows
unsatisfactory slow convergence rates for transport dominated problems. While
this has triggered a large amount of work on nonlinear model reduction
techniques, we are lacking a benchmark to evaluate their optimal performance.
  Nonlinear benchmarks like manifold/stable/Lipschitz width applied to the
solution manifold are often trivial if the degrees of freedom exceed the
parameter dimension and ignore desirable structure as offline/online
decompositions. In this paper, we show that the same benchmarks applied to the
full reduced order model pipeline from PDE to parametric quantity of interest
provide non-trivial benchmarks and we prove lower bounds for transport
equations.
