A Novel Neural Network-Based Approach Comparable to High-Precision Finite Difference Methods
Deep learning methods using neural networks for solving partial differential equations (PDEs) have emerged as a new paradigm. However, many of these methods approximate solutions by optimizing loss functions, often encountering convergence issues and accuracy limitations. In this paper, we propose a...
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2025-01-01
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| author | Fanghua Pei Fujun Cao Yongbin Ge |
| author_facet | Fanghua Pei Fujun Cao Yongbin Ge |
| author_sort | Fanghua Pei |
| collection | DOAJ |
| description | Deep learning methods using neural networks for solving partial differential equations (PDEs) have emerged as a new paradigm. However, many of these methods approximate solutions by optimizing loss functions, often encountering convergence issues and accuracy limitations. In this paper, we propose a novel deep learning approach that leverages the expressive power of neural networks to generate basis functions. These basis functions are then used to create trial solutions, which are optimized using the least-squares method to solve for coefficients in a system of linear equations. This method integrates the strengths of streaming PINNs and the traditional least-squares method, offering both flexibility and a high accuracy. We conducted numerical experiments to compare our method with the results of high-order finite difference schemes and several commonly used neural network methods (PINNs, lbPINNs, ELMs, and PIELMs). Thanks to the mesh-less feature of the neural network, it is particularly effective for complex geometries. The numerical results demonstrate that our method significantly enhances the accuracy of deep learning in solving PDEs, achieving error levels comparable to high-accuracy finite difference methods. |
| format | Article |
| id | doaj-art-7428b2ca72cb465fb95fa62cf500ba9f |
| institution | Kabale University |
| issn | 2075-1680 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | MDPI AG |
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| series | Axioms |
| spelling | doaj-art-7428b2ca72cb465fb95fa62cf500ba9f2025-01-24T13:22:21ZengMDPI AGAxioms2075-16802025-01-011417510.3390/axioms14010075A Novel Neural Network-Based Approach Comparable to High-Precision Finite Difference MethodsFanghua Pei0Fujun Cao1Yongbin Ge2School of Mathematical Statistics, Ningxia University, Yinchuan 750021, ChinaSchool of Science, Inner Mongolia University of Science and Technology, Baotou 014010, ChinaSchool of Mathematical Statistics, Ningxia University, Yinchuan 750021, ChinaDeep learning methods using neural networks for solving partial differential equations (PDEs) have emerged as a new paradigm. However, many of these methods approximate solutions by optimizing loss functions, often encountering convergence issues and accuracy limitations. In this paper, we propose a novel deep learning approach that leverages the expressive power of neural networks to generate basis functions. These basis functions are then used to create trial solutions, which are optimized using the least-squares method to solve for coefficients in a system of linear equations. This method integrates the strengths of streaming PINNs and the traditional least-squares method, offering both flexibility and a high accuracy. We conducted numerical experiments to compare our method with the results of high-order finite difference schemes and several commonly used neural network methods (PINNs, lbPINNs, ELMs, and PIELMs). Thanks to the mesh-less feature of the neural network, it is particularly effective for complex geometries. The numerical results demonstrate that our method significantly enhances the accuracy of deep learning in solving PDEs, achieving error levels comparable to high-accuracy finite difference methods.https://www.mdpi.com/2075-1680/14/1/75neural networkhigh-order compact finite difference methoddeep learningPDEs |
| spellingShingle | Fanghua Pei Fujun Cao Yongbin Ge A Novel Neural Network-Based Approach Comparable to High-Precision Finite Difference Methods Axioms neural network high-order compact finite difference method deep learning PDEs |
| title | A Novel Neural Network-Based Approach Comparable to High-Precision Finite Difference Methods |
| title_full | A Novel Neural Network-Based Approach Comparable to High-Precision Finite Difference Methods |
| title_fullStr | A Novel Neural Network-Based Approach Comparable to High-Precision Finite Difference Methods |
| title_full_unstemmed | A Novel Neural Network-Based Approach Comparable to High-Precision Finite Difference Methods |
| title_short | A Novel Neural Network-Based Approach Comparable to High-Precision Finite Difference Methods |
| title_sort | novel neural network based approach comparable to high precision finite difference methods |
| topic | neural network high-order compact finite difference method deep learning PDEs |
| url | https://www.mdpi.com/2075-1680/14/1/75 |
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