  White dwarfs (WDs) embedded in gaseous disks of active galactic nucleus
(AGNs) can rapidly accrete materials from the disks and grow in mass to reach
or even exceed the Chandrasekhar limit. Binary WD (BWD) mergers are also
believed to occur in AGN accretion disks. We study observational signatures
from these events. We suggest that mass-accreting WDs and BWD mergers in AGN
disks can lead to thermonuclear explosions that drive an ejecta shock breakout
from the disk surface and power a slow-rising, relatively dim Type Ia supernova
(SN). Such SNe Ia may be always outshone by the emission of the AGN disk around
the supermassive black hole (BH) with a mass of $M_{\rm SMBH}\gtrsim
10^8\,M_\odot$. Besides, accretion-induced collapses (AICs) of WDs in AGN disks
may occur sometimes, which may form highly-magnetized millisecond neutron stars
(NSs). The subsequent spin-down process of this nascent magnetar can deposit
its rotational energy into the disk materials, resulting in a magnetar-driven
shock breakout and a luminous magnetar-powered transient. We show that such an
AIC event could power a rapidly evolving and luminous transient for a magnetic
field of $B\sim10^{15}\,{\rm G}$. The rising time and peak luminosity of the
transient, powered by a magnetar with $B\sim10^{14}\,{\rm G}$, are predicted to
have similar properties with those of superluminous supernovae. AIC events
taking place in the inner parts of the disk around a relatively less massive
supermassive BHs ($M_{\rm SMBH}\lesssim10^8\,M_\odot$) are more likely to power
the transients that are much brighter than the AGN disk emission and hence
easily to be identified.
