DIMON: Learning Solution Operators of Partial Differential Equations on
  a Diffeomorphic Family of Domains
Minglang Yin Nicolas Charon Ryan Brody Lu Lu Natalia Trayanova Mauro Maggioni
The solution of a PDE over varying initial/boundary conditions on multiple
domains is needed in a wide variety of applications, but it is computationally
expensive if the solution is computed de novo whenever the initial/boundary
conditions of the domain change. We introduce a general operator learning
framework, called DIffeomorphic Mapping Operator learNing (DIMON) to learn
approximate PDE solutions over a family of domains $\{\Omega_{\theta}}_\theta$,
that learns the map from initial/boundary conditions and domain $\Omega_\theta$
to the solution of the PDE, or to specified functionals thereof. DIMON is based
on transporting a given problem (initial/boundary conditions and domain
$\Omega_{\theta}$) to a problem on a reference domain $\Omega_{0}$, where
training data from multiple problems is used to learn the map to the solution
on $\Omega_{0}$, which is then re-mapped to the original domain
$\Omega_{\theta}$. We consider several problems to demonstrate the performance
of the framework in learning both static and time-dependent PDEs on non-rigid
geometries; these include solving the Laplace equation, reaction-diffusion
equations, and a multiscale PDE that characterizes the electrical propagation
on the left ventricle. This work paves the way toward the fast prediction of
PDE solutions on a family of domains and the application of neural operators in
engineering and precision medicine.