  This thesis work represents the first efforts to combine population synthesis
studies of the Galactic isolated neutron stars with deep-learning techniques
with the aim of better understanding neutron-star birth properties and
evolution. In particular, we develop a flexible population-synthesis framework
to model the dynamical and magneto-rotational evolution of neutron stars, their
emission in radio and their detection with radio telescopes. We first study the
feasibility of using deep neural networks to infer the dynamical properties at
birth and then explore a simulation-based inference approach to predict the
birth magnetic-field and spin-period distributions and the late-time
magnetic-field decay for the observed radio pulsar population. Our results for
the birth magneto-rotational properties agree with the findings of previous
works while we constrain the late-time evolution of the magnetic field in
neutron stars for the first time. Moreover, this thesis also studies possible
scenarios to explain the puzzling nature of recently discovered periodic radio
sources with very long periods of the order of thousands of seconds. In
particular, by assuming a neutron-star origin, we study the spin-period
evolution of a newborn neutron star interacting with a supernova fallback disk
and find that the combination of strong, magnetar-like magnetic fields and
moderate accretion rates can lead to very large spin periods on timescales of
ten thousands of years. Moreover, we perform population synthesis studies to
assess the possibility for these sources to be either neutron stars or magnetic
white dwarfs emitting coherently through magnetic dipolar losses. These
discoveries have opened up a new perspective on the neutron-star population and
have started to question our current understanding of how coherent radio
emission is produced in pulsar magnetospheres.
