Saved in:
| Main Authors: | , , , , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2402.07250 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866929240819105792 |
|---|---|
| author | Yin, Minglang Charon, Nicolas Brody, Ryan Lu, Lu Trayanova, Natalia Maggioni, Mauro |
| author_facet | Yin, Minglang Charon, Nicolas Brody, Ryan Lu, Lu Trayanova, Natalia Maggioni, Mauro |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_07250 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | DIMON: Learning Solution Operators of Partial Differential Equations on a Diffeomorphic Family of Domains Yin, Minglang Charon, Nicolas Brody, Ryan Lu, Lu Trayanova, Natalia Maggioni, Mauro Machine Learning Artificial Intelligence Computational Engineering, Finance, and Science 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. |
| title | DIMON: Learning Solution Operators of Partial Differential Equations on a Diffeomorphic Family of Domains |
| topic | Machine Learning Artificial Intelligence Computational Engineering, Finance, and Science |
| url | https://arxiv.org/abs/2402.07250 |