  We propose a data-driven model order reduction (MOR) technique for
parametrized partial differential equations that exhibit parameter-dependent
jump-discontinuities. Such problems have poor-approximability in a linear space
and therefore, are challenging for standard MOR techniques. We build upon the
methodology of approximating the map between the parameter domain and the
expansion coefficients of the reduced basis via regression. The online stage
queries the regression model for the expansion coefficients and recovers a
reduced approximation for the solution. We propose to apply this technique to a
transformed solution that results from composing the solution with a spatial
transform. Unlike the (untransformed) solution, it is sufficiently regular
along the parameter domain and thus, is well-approximable in a low-dimensional
linear space. To recover an approximation for the (untransformed) solution, we
propose an online efficient regression-based technique that approximates the
inverse of the spatial transform. Our method features a decoupled online and
offline stage, and benchmark problems involving hyperbolic and parabolic
equations demonstrate its effectiveness.
