  We have constructed deep neural networks, which can map fluctuating
photo-electron spectra obtained from noisy pulses to spectra from noise-free
pulses. The network is trained on spectra from noisy pulses in combination with
random Hamilton matrices, representing systems which could exist but do not
necessarily exist. In [Giri et al., Phys. Rev. Lett. 124,113201 (2020)] we
performed a purification of fluctuating spectra, that is mapping them to those
from Fourier-limited Gaussian pulses. Here, we investigate the performance of
such neural-network-based maps for predicting spectra of double pulses, pulses
with a chirp and even partially-coherent pulses pulses from fluctuating spectra
generated by noisy pulses. Secondly, we demonstrate that along with a
purification of a fluctuating double-pulse spectrum, one can estimate the
time-delay of the underlying double pulse, an attractive feature for
single-shot spectra from SASE FELs. We demonstrate our approach with resonant
two-photon ionization, a non-linear process, sensitive to details of the laser
pulse.
