Chebyshev Spectral Neural Networks for Solving Partial Differential Equations
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , , |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866913416288927744 |
|---|---|
| author | Yin, Pengsong Ling, Shuo Ying, Wenjun |
| author_facet | Yin, Pengsong Ling, Shuo Ying, Wenjun |
| contents | The purpose of this study is to utilize the Chebyshev spectral method neural network(CSNN) model to solve differential equations. This approach employs a single-layer neural network wherein Chebyshev spectral methods are used to construct neurons satisfying boundary conditions. The study uses a feedforward neural network model and error backpropagation principles, utilizing automatic differentiation (AD) to compute the loss function. This method avoids the need to solve non-sparse linear systems, making it convenient for algorithm implementation and solving high-dimensional problems. The unique sampling method and neuron architecture significantly enhance the training efficiency and accuracy of the neural network. Furthermore, multiple networks enables the Chebyshev spectral method to handle equations on more complex domains. The numerical efficiency and accuracy of the CSNN model are investigated through testing on elliptic partial differential equations, and it is compared with the well-known Physics-Informed Neural Network(PINN) method. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_03347 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Chebyshev Spectral Neural Networks for Solving Partial Differential Equations Yin, Pengsong Ling, Shuo Ying, Wenjun Numerical Analysis Machine Learning Mathematical Physics The purpose of this study is to utilize the Chebyshev spectral method neural network(CSNN) model to solve differential equations. This approach employs a single-layer neural network wherein Chebyshev spectral methods are used to construct neurons satisfying boundary conditions. The study uses a feedforward neural network model and error backpropagation principles, utilizing automatic differentiation (AD) to compute the loss function. This method avoids the need to solve non-sparse linear systems, making it convenient for algorithm implementation and solving high-dimensional problems. The unique sampling method and neuron architecture significantly enhance the training efficiency and accuracy of the neural network. Furthermore, multiple networks enables the Chebyshev spectral method to handle equations on more complex domains. The numerical efficiency and accuracy of the CSNN model are investigated through testing on elliptic partial differential equations, and it is compared with the well-known Physics-Informed Neural Network(PINN) method. |
| title | Chebyshev Spectral Neural Networks for Solving Partial Differential Equations |
| topic | Numerical Analysis Machine Learning Mathematical Physics |
| url | https://arxiv.org/abs/2407.03347 |