Learning the boundary-to-domain mapping using Lifting Product Fourier Neural Operators for partial differential equations
Fuente:
arXiv
Salvato in:
| Autori principali: | , , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866916307988905984 |
|---|---|
| author | Kashi, Aditya Daw, Arka Meena, Muralikrishnan Gopalakrishnan Lu, Hao |
| author_facet | Kashi, Aditya Daw, Arka Meena, Muralikrishnan Gopalakrishnan Lu, Hao |
| contents | Neural operators such as the Fourier Neural Operator (FNO) have been shown to provide resolution-independent deep learning models that can learn mappings between function spaces. For example, an initial condition can be mapped to the solution of a partial differential equation (PDE) at a future time-step using a neural operator. Despite the popularity of neural operators, their use to predict solution functions over a domain given only data over the boundary (such as a spatially varying Dirichlet boundary condition) remains unexplored. In this paper, we refer to such problems as boundary-to-domain problems; they have a wide range of applications in areas such as fluid mechanics, solid mechanics, heat transfer etc. We present a novel FNO-based architecture, named Lifting Product FNO (or LP-FNO) which can map arbitrary boundary functions defined on the lower-dimensional boundary to a solution in the entire domain. Specifically, two FNOs defined on the lower-dimensional boundary are lifted into the higher dimensional domain using our proposed lifting product layer. We demonstrate the efficacy and resolution independence of the proposed LP-FNO for the 2D Poisson equation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_16740 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Learning the boundary-to-domain mapping using Lifting Product Fourier Neural Operators for partial differential equations Kashi, Aditya Daw, Arka Meena, Muralikrishnan Gopalakrishnan Lu, Hao Machine Learning Numerical Analysis 65N99, 68T07 I.2.1; J.2 Neural operators such as the Fourier Neural Operator (FNO) have been shown to provide resolution-independent deep learning models that can learn mappings between function spaces. For example, an initial condition can be mapped to the solution of a partial differential equation (PDE) at a future time-step using a neural operator. Despite the popularity of neural operators, their use to predict solution functions over a domain given only data over the boundary (such as a spatially varying Dirichlet boundary condition) remains unexplored. In this paper, we refer to such problems as boundary-to-domain problems; they have a wide range of applications in areas such as fluid mechanics, solid mechanics, heat transfer etc. We present a novel FNO-based architecture, named Lifting Product FNO (or LP-FNO) which can map arbitrary boundary functions defined on the lower-dimensional boundary to a solution in the entire domain. Specifically, two FNOs defined on the lower-dimensional boundary are lifted into the higher dimensional domain using our proposed lifting product layer. We demonstrate the efficacy and resolution independence of the proposed LP-FNO for the 2D Poisson equation. |
| title | Learning the boundary-to-domain mapping using Lifting Product Fourier Neural Operators for partial differential equations |
| topic | Machine Learning Numerical Analysis 65N99, 68T07 I.2.1; J.2 |
| url | https://arxiv.org/abs/2406.16740 |