  Since the turn of the century, astronomers have been exploiting the rich
information afforded by combining stellar kinematic maps and imaging in an
attempt to recover the intrinsic, three-dimensional (3D) shape of a galaxy. A
common intrinsic shape recovery method relies on an expected monotonic
relationship between the intrinsic misalignment of the kinematic and
morphological axes and the triaxiality parameter. Recent studies have, however,
cast doubt about underlying assumptions relating shape and intrinsic kinematic
misalignment. In this work, we aim to recover the 3D shape of individual
galaxies using their projected stellar kinematic and flux distributions using a
supervised machine learning approach with mixture density network (MDN). Using
a mock dataset of the EAGLE hydrodynamical cosmological simulation, we train
the MDN model for a carefully selected set of common kinematic and photometric
parameters. Compared to previous methods, we demonstrate potential improvements
achieved with the MDN model to retrieve the 3D galaxy shape along with the
uncertainties, especially for prolate and triaxial systems. We make specific
recommendations for recovering galaxy intrinsic shapes relevant for current and
future integral field spectroscopic galaxy surveys.
