We have introduced the idea of synthetic Hamilton matrices (SHMs), random representations for the dynamics of systems coupled to light, which could exist but not necessarily exist in nature. This approach enables sampling of the training space solving the dynamics with SHMs efficiently incorporating sufficiently generic features to be transferable to real systems. They serve the purpose to augment training data for DL with DNNs. We demonstrated that this SHM-DL approach works by purifying photoelectron spectra from noisy pulses and identifying pulse delays which jitter in an unknown manner, as supplied by X-ray Free-electron Lasers. The approach is physics-oriented and therefore promises physics insight beyond the prediction of spectra through the DL based identification of the relevant Hamilton matrix from a spectrum for a system, unknown to the DNN.