A Review of Neural Network Solvers for Second-order Boundary Value Problems
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866913439022055424 |
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| author | Sau, Ramesh Chandra Yin, Luowei |
| author_facet | Sau, Ramesh Chandra Yin, Luowei |
| contents | Deep learning-based partial differential equation(PDE) solvers have received much attention in the past few years. Methods of this category can solve a wide range of PDEs with high accuracy, typically by transforming the problems into highly nonlinear optimization problems of neural network parameters. This work reviews several deep learning solvers proposed a few years ago, including PINN, WAN, DRM, and VPINN. Numerical results are provided to make comparisons amongst them and address the importance of loss formulation and the optimization method. A rigorous error analysis for PINN is also presented. Finally, we discuss the current limitations and bottlenecks of these methods. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_00442 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Review of Neural Network Solvers for Second-order Boundary Value Problems Sau, Ramesh Chandra Yin, Luowei Numerical Analysis Deep learning-based partial differential equation(PDE) solvers have received much attention in the past few years. Methods of this category can solve a wide range of PDEs with high accuracy, typically by transforming the problems into highly nonlinear optimization problems of neural network parameters. This work reviews several deep learning solvers proposed a few years ago, including PINN, WAN, DRM, and VPINN. Numerical results are provided to make comparisons amongst them and address the importance of loss formulation and the optimization method. A rigorous error analysis for PINN is also presented. Finally, we discuss the current limitations and bottlenecks of these methods. |
| title | A Review of Neural Network Solvers for Second-order Boundary Value Problems |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2407.00442 |