Advection dominated problems represent still nowadays a great challenge for the Model Order Reduction community, because of their intrinsic difficult nature. In this talk we will focus on hyperbolic problems with self-similar solutions. I will present a MOR approach for transport dominated problems, in the non-parametrized and in the parametrized setting, with a particular focus on the SOD problem in 1D, the DMR problem and the triple point problem in 2D. The approach is based on the definition of suitable deformation maps from the physical domain into itself: these maps are obtained by means of an optimization procedure. Once the map is found, a standard POD on the modified snapshots is performed. For the online phase, an Artificial Neural Network approach is used to compute the coefficients of the online solution. The whole procedure represents a first step towards an ALE approach, and is applied to problems where the solution presents multiple travelling discontinuities (shocks, rarefactions), whose location in the physical domain is unknown. Promising results are shown, to highlight the good performance of the whole methodology.