In this work we propose an Uncertainty Quantification methodology for the evolution of sedimentary basins undergoing mechanical and geochemical compaction processes, which we model as a coupled, time-dependent, non-linear, monodimensional (depth-only) system of PDEs with uncertain parameters. Specifically, we consider multi-layered basins, in which each layer is characterized by a different material. The multi-layered structure gives rise to discontinuities in the dependence of the state variables on the uncertain parameters. Because of these discontinuites, an appropriate treatment is needed for surrogate modeling techniques such as sparse grids to be effective. To this end, we propose a two-steps methodology which relies on a change of coordinate system to align the discontinuities of the target function within the random parameter space. Once this alignement has been computed, a standard sparse grid approximation of the state variables can be performed. The effectiveness of this procedure is due to the fact that the physical locations of the interfaces among layers feature a smooth dependence on the random parameters and are therefore amenable to sparse grid polynomial approximations. We showcase the capabilities of our numerical methodologies through some synthetic test cases. References: Ivo Colombo, Fabio Nobile, Giovanni Porta, Anna Scotti, Lorenzo Tamellini, Uncertainty Quantification of geochemical and mechanical compaction in layered sedimentary basins, Computer Methods in Applied Mechanics and Engineering.