DIMON: Learning Solution Operators of Partial Differential Equations on a Diffeomorphic Family of Domains
This work addresses the need for efficient PDE solutions across multiple domains in engineering and precision medicine, representing a novel method for a known bottleneck rather than an incremental improvement.
The authors tackled the problem of solving PDEs on varying domains, which is computationally expensive when done from scratch for each domain change, by introducing DIMON, a framework that learns solution operators and achieves fast prediction with demonstrated performance on static and time-dependent PDEs like the Laplace equation and reaction-diffusion equations, including applications in precision medicine.
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 $\{Ω_θ}_θ$, that learns the map from initial/boundary conditions and domain $Ω_θ$ to the solution of the PDE, or to specified functionals thereof. DIMON is based on transporting a given problem (initial/boundary conditions and domain $Ω_θ$) to a problem on a reference domain $Ω_{0}$, where training data from multiple problems is used to learn the map to the solution on $Ω_{0}$, which is then re-mapped to the original domain $Ω_θ$. 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.