  Gravitational wave observations of the inspiral of stellar-mass compact
objects into massive black holes (MBHs), extreme mass ratio inspirals (EMRIs),
enable precision measurements of parameters such as the MBH mass and spin. The
Laser Interferometer Space Antenna is expected to detect sufficient EMRIs to
probe the underlying source population, testing theories of the formation and
evolution of MBHs and their environments. Population studies are subject to
selection effects that vary across the EMRI parameter space, which bias
inference results if unaccounted for. This bias can be corrected, but
evaluating the detectability of many EMRI signals is computationally expensive.
We mitigate this cost by (i) constructing a rapid and accurate neural network
interpolator capable of predicting the signal-to-noise ratio of an EMRI from
its parameters, and (ii) further accelerating detectability estimation with a
neural network that learns the selection function, leveraging our first neural
network for data generation. The resulting framework rapidly estimates the
selection function, enabling a full treatment of EMRI detectability in
population inference analyses. We apply our method to an astrophysically
motivated EMRI population model, demonstrating the potential selection biases
and subsequently correcting for them. Accounting for selection effects, we
predict that LISA will measure the MBH mass function slope to a precision of
8.8%, the CO mass function slope to a precision of 4.6%, the width of the MBH
spin magnitude distribution to a precision of 10% and the event rate to a
precision of 12% with EMRIs at redshifts below z=6.
