  Representing physical signals at different scales is among the most
challenging problems in engineering. Several multi-scale modeling tools have
been developed to describe physical systems governed by \emph{Partial
Differential Equations} (PDEs). These tools are at the crossroad of principled
physical models and numerical schema. Recently, data-driven models have been
introduced to speed-up the approximation of PDE solutions compared to numerical
solvers. Among these recent data-driven methods, neural integral operators are
a class that learn a mapping between function spaces. These functions are
discretized on graphs (meshes) which are appropriate for modeling interactions
in physical phenomena. In this work, we study three multi-resolution schema
with integral kernel operators that can be approximated with \emph{Message
Passing Graph Neural Networks} (MPGNNs). To validate our study, we make
extensive MPGNNs experiments with well-chosen metrics considering steady and
unsteady PDEs.
