  We present a registration method for model reduction of parametric partial
differential equations with dominating advection effects and moving features.
Registration refers to the use of a parameter-dependent mapping to make the set
of solutions to these equations more amicable for approximation using classical
reduced basis methods. The proposed approach utilizes concepts from optimal
transport theory, as we utilize Monge embeddings to construct these mappings in
a purely data-driven way. The method relies on one interpretable
hyper-parameter. We discuss how our approach relates to existing works that
combine model order reduction and optimal transport theory. Numerical results
are provided to demonstrate the effect of the registration. This includes a
model problem where the solution is itself a probability density and one where
it is not.
