Reduced Order Modeling of Partial Differential Equations on Parameter-Dependent Domains Using Deep Neural Networks

Abstract

Partial differential equations (PDEs) are widely used for modeling various physical phenomena. These equations often depend on certain parameters, necessitating either the identification of optimal parameters or the solution of the equations over multiple parameters. 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 built using a set of precomputed solutions (snapshots) corresponding to different parameter values. Recently, Deep Learning ROMs (DL-ROMs) have been introduced as a new method to obtain ROM, offering improved flexibility and performance. In many cases, the domain on which the PDE is defined also varies. Capturing this variation is important for building accurate ROMs but is often difficult, especially when the domain has a complex structure or changes topology. In this paper, we propose a Deep-ROM framework that can automatically extract useful domain parametrization and incorporate it into the model. Unlike traditional domain parameterization methods, our approach does not require user-defined control points and can effectively handle domains with varying numbers of components. It can also learn from domain data even when no mesh is available. Using deep autoencoders, our approach reduces the dimensionality of both the PDE solution and the domain representation, making it possible to approximate solutions efficiently across a wide range of domain shapes and parameter values. We demonstrate that our approach produces parametrizations that yield solution accuracy comparable to models using exact parameters. Importantly, our model remains stable under moderate geometric variations in the domain, such as boundary deformations and noise - scenarios where traditional ROMs often require remeshing or manual adjustment.