  Given a set of solution snapshots of a hyperbolic PDE, we are interested in
learning a reduced order model (ROM). To this end, we propose a novel decompose
then learn approach. We decompose the solution by expressing it as a
composition of a transformed solution and a de-transformer. Our idea is to
learn a ROM for both these objects, which, unlike the solution, are well
approximable in a linear reduced space. A ROM for the (untransformed) solution
is then recovered via a recomposition. The transformed solution results from
composing the solution with a spatial transform that aligns the spatial
discontinuities. Furthermore, the de-transformer is the inverse of the spatial
transform and lets us recover a ROM for the solution. We consider an image
registration technique to compute the spatial transform, and to learn a ROM, we
resort to the dynamic mode decomposition (DMD) methodology. Several benchmark
problems demonstrate the effectiveness our method in representing the data and
as a predictive tool.
