  Accretion disks of active galactic nuclei (AGN) have been proposed as
promising sites for producing both (stellar-mass) compact object mergers and
extreme mass ratio inspirals. Along with the disk-assisted migration/evolution
process, ambient gas materials inevitably accrete onto the compact objects. The
description of this process is subject to significant theoretical uncertainties
in previous studies. It was commonly assumed that either an Eddington accretion
rate or a Bondi accretion rate (or any rate in between) takes place, although
these two rates can differ from each other by several orders of magnitude. As a
result, the mass and spin evolution of compact objects within AGN disks are
essentially unknown. In this work, we construct a relativistic supercritical
inflow-outflow model for black hole (BH) accretion. We show that the radiation
efficiency of the supercritical accretion of a stellar-mass BH (sBH) is
generally too low to explain the proposed electromagnetic counterpart of
GW190521. Applying this model to sBHs embedded in AGN disks, we find that,
although the gas inflow rates at Bondi radii of these sBHs are in general
highly super-Eddington, a large fraction of inflowing gas eventually escapes as
outflows so that only a small fraction accretes onto the sBH, resulting in
mildly super-Eddington BH absorption in most cases. We also implement this
inflow-outflow model to study the evolution of neutron stars (NS) and white
dwarfs (WD) in AGN disks, taking into account corrections from star sizes and
star magnetic fields. It turns out to be difficult for WDs to grow to the
Chandrasekhar limit via accretion because WDs are spun up more efficiently to
reach the shedding limit before the Chandrasekhar limit. For NSs the
accretion-induced collapse is possible if NS magnetic fields are sufficiently
strong, keeping the NS in a slow rotation state during accretion.
