  Using neural networks, we integrate the ability to account for Doppler
smearing due to a pulsar's orbital motion with the pulsar population synthesis
package \psrpoppy\ to develop accurate modeling of the observed binary pulsar
population. As a first application, we show that binary neutron star systems
where the two components have highly unequal mass are, on average, easier to
detect than systems which are symmetric in mass. We then investigate the
population of ultra-compact ($1.5 \, {\rm min} \leq P_{\rm b} \leq 15\,\rm
min$) neutron star--white dwarf (NS--WD) and double neutron star (DNS) systems
which are promising sources for the Laser Interferometer Space Antenna
gravitational-wave detector. Given the non-detection of these systems in radio
surveys thus far, we estimate a 95\% confidence upper limit of $\sim$1450 and
$\sim$1100 ultra-compact NS--WD and DNS systems in the Milky Way that are
beaming towards the Earth respectively. We also show that using survey
integration times in the range 20~s to 200~s with time-domain resampling will
maximize the signal-to-noise ratio as well as the probability of detection of
these ultra-compact binary systems. Among all the large scale radio pulsar
surveys, those that are currently being carried out at the Arecibo radio
telescope have $\sim$50--80\% chance of detecting at least one of these systems
using current integration integration times and $\sim$80--95\% using optimal
integration times in the next several years.
