  Neutrinos can undergo fast flavor conversions (FFCs) within extremely dense
astrophysical environments such as core-collapse supernovae (CCSNe) and neutron
star mergers (NSMs). In this study, we explore FFCs in a \emph{multi-energy}
neutrino gas, revealing that when the FFC growth rate significantly exceeds
that of the vacuum Hamiltonian, all neutrinos (regardless of energy) share a
common survival probability dictated by the energy-integrated neutrino
spectrum. We then employ physics-informed neural networks (PINNs) to predict
the asymptotic outcomes of FFCs within such a multi-energy neutrino gas. These
predictions are based on the first two moments of neutrino angular
distributions for each energy bin, typically available in state-of-the-art CCSN
and NSM simulations. Our PINNs achieve errors as low as $\lesssim6\%$ and
$\lesssim 18\%$ for predicting the number of neutrinos in the electron channel
and the relative absolute error in the neutrino moments, respectively.
