  $\beta$-$\mathrm{Ga}_{2}\mathrm{O}_{3}$ has gained intensive interests of
research and application as an ultrawide bandgap semiconductor. Epitaxial
growth technique of the $\beta$-$\mathrm{Ga}_{2}\mathrm{O}_{3}$ thin film
possesses a fundamental and vital role in the
$\mathrm{Ga}_{2}\mathrm{O}_{3}$-based device fabrication. In this work,
epitaxial growth mechanisms of $\beta$-$\mathrm{Ga}_{2}\mathrm{O}_{3}$ with
four low Miller-index facets, namely (100), (010), (001), and
($\overline{2}$01), are systematically explored using large-scale
machine-learning molecular dynamics simulations at the atomic scale. The
simulations reveal that the migration of the face-centered cubic stacking O
sublattice plays a predominant role in rationalizing the different growth
mechanisms between (100)/(010)/(001) and ($\overline{2}$01) orientations. The
resultant complex combinations of the stacking faults and twin boundaries are
carefully identified, and shows a good agreement with the experimental
observation and ab initio calculation. Our results provide useful insights into
the gas-phase epitaxial growth of the $\beta$-$\mathrm{Ga}_{2}\mathrm{O}_{3}$
thin films and suggest possible ways to tailor its properties for specific
applications.
