Discontinuous hybrid neural networks for the one-dimensional partial differential equations
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866913838679457792 |
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| author | Wang, Xiaoyu Yuan, Long Yu, Yao |
| author_facet | Wang, Xiaoyu Yuan, Long Yu, Yao |
| contents | A feedforward neural network, including hidden layers, motivated by nonlinear functions (such as Tanh, ReLU, and Sigmoid functions), exhibits uniform approximation properties in Sobolev space, and discontinuous neural networks can reduce computational complexity. In this work, we present a discontinuous hybrid neural network method for solving the partial differential equations, construct a new hybrid loss functional that incorporates the variational of the approximation equation, interface jump stencil and boundary constraints. The RMSprop algorithm and discontinuous Galerkin method are employed to update the nonlinear parameters and linear parameters in neural networks, respectively. This approach guarantees the convergence of the loss functional and provides an approximate solution with high accuracy. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_09911 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Discontinuous hybrid neural networks for the one-dimensional partial differential equations Wang, Xiaoyu Yuan, Long Yu, Yao Numerical Analysis 65N30, 65N55, 68T07 A feedforward neural network, including hidden layers, motivated by nonlinear functions (such as Tanh, ReLU, and Sigmoid functions), exhibits uniform approximation properties in Sobolev space, and discontinuous neural networks can reduce computational complexity. In this work, we present a discontinuous hybrid neural network method for solving the partial differential equations, construct a new hybrid loss functional that incorporates the variational of the approximation equation, interface jump stencil and boundary constraints. The RMSprop algorithm and discontinuous Galerkin method are employed to update the nonlinear parameters and linear parameters in neural networks, respectively. This approach guarantees the convergence of the loss functional and provides an approximate solution with high accuracy. |
| title | Discontinuous hybrid neural networks for the one-dimensional partial differential equations |
| topic | Numerical Analysis 65N30, 65N55, 68T07 |
| url | https://arxiv.org/abs/2505.09911 |