Using neural networks, we integrate the ability to account for Doppler smearing due to a pulsars 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.
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