DIMON: Learning Solution Operators of Partial Differential Equations on a Diffeomorphic Family of Domains

• DOI: 10.48550/arxiv.2402.07250
• Publication Date: 2024-02-11
• Authors: Minglang Yin; Nicolas Charon; Ryan Brody; Lu Lu; Natalia A. Trayanova; Mauro Maggioni
• Keywords: Diffeomorphism

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