  Model order reduction (MOR) techniques have always struggled in compressing
information for advection dominated problems. Their linear nature does not
allow to accelerate the slow decay of the Kolmogorov $N$--width of these
problems. In the last years, new nonlinear algorithms obtained smaller reduced
spaces. In these works only the offline phase of these algorithms was shown. In
this work, we study MOR algorithms for unsteady parametric advection dominated
hyperbolic problems, giving a complete offline and online description and
showing the time saving in the online phase. We propose an arbitrary
Lagrangian--Eulerian approach that modifies both the offline and online phases
of the MOR process. This allows to calibrate the advected features on the same
position and to strongly compress the reduced spaces. The basic MOR algorithms
used are the classical Greedy, EIM and POD, while the calibration map is
learned through polynomial regression and artificial neural networks. In the
performed simulations we show how the new algorithm defeats the classical
method on many equations with nonlinear fluxes and with different boundary
conditions. Finally, we compare the results obtained with different calibration
maps.
