DIMON: Learning Solution Operators of Partial Differential Equations on a Diffeomorphic Family of Domains
<|reference_start|>DIMON: Learning Solution Operators of Partial Differential Equations on a Diffeomorphic Family of Domains: 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.<|reference_end|>
@article{yin2024dimon:,
    title={DIMON: Learning Solution Operators of Partial Differential Equations on
  a Diffeomorphic Family of Domains},
    author={Minglang Yin, Nicolas Charon, Ryan Brody, Lu Lu, Natalia Trayanova,
  Mauro Maggioni},
    journal={arXiv preprint arXiv:2402.07250},
    year={2024},
    archivePrefix={arXiv},
    eprint={2402.07250},
    primaryClass={cs.LG cs.AI cs.CE}
}