  Partial differential equations (PDEs) are extensively utilized for modeling
various physical phenomena. These equations often depend on certain parameters,
necessitating either the identification of optimal parameters or solving the
equation across multiple parameters to understand how a structure might react
under different conditions. Performing an exhaustive search over the parameter
space requires solving the PDE multiple times, which is generally impractical.
To address this challenge, Reduced Order Models (ROMs) are constructed from a
snapshot dataset comprising parameter-solution pairs. ROMs facilitate the rapid
solving of PDEs for new parameters.
  Recently, Deep Learning ROMs (DL-ROMs) have been introduced as a new method
to obtain ROM. Additionally, the PDE of interest may depend on the domain,
which can be characterized by parameters or measurements and may evolve with
the system, requiring parametrization for ROM construction. In this paper, we
develop a Deep-ROM capable of extracting and efficiently utilizing domain
parametrization. Unlike traditional domain parametrization methods, our
approach does not require user-defined control points and can effectively
handle domains with varying numbers of components. Moreover, our model can
derive meaningful parametrization even when a domain mesh is unavailable, a
common scenario in biomedical applications. Our work leverages Deep Neural
Networks to effectively reduce the dimensionality of the PDE and the domain
characteristic function.
