A Novel Neural Network-Based Approach Comparable to High-Precision Finite Difference Methods
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| Xuất bản năm: | Axioms vol. 14, no. 1 (2025), p. 75 |
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| Tác giả chính: | |
| Tác giả khác: | , |
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MDPI AG
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| Truy cập trực tuyến: | Citation/Abstract Full Text + Graphics Full Text - PDF |
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| 022 | |a 2075-1680 | ||
| 024 | 7 | |a 10.3390/axioms14010075 |2 doi | |
| 035 | |a 3159334398 | ||
| 045 | 2 | |b d20250101 |b d20251231 | |
| 084 | |a 231430 |2 nlm | ||
| 100 | 1 | |a Pei, Fanghua |u School of Mathematical Statistics, Ningxia University, Yinchuan 750021, China; <email>371102a29nj.cdb@sin.cn</email> | |
| 245 | 1 | |a A Novel Neural Network-Based Approach Comparable to High-Precision Finite Difference Methods | |
| 260 | |b MDPI AG |c 2025 | ||
| 513 | |a Journal Article | ||
| 520 | 3 | |a 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. | |
| 653 | |a Finite volume method | ||
| 653 | |a Accuracy | ||
| 653 | |a Physics | ||
| 653 | |a Partial differential equations | ||
| 653 | |a Deep learning | ||
| 653 | |a Mathematical analysis | ||
| 653 | |a Neural networks | ||
| 653 | |a Adaptability | ||
| 653 | |a Mathematical models | ||
| 653 | |a Finite difference method | ||
| 653 | |a Least squares method | ||
| 653 | |a Basis functions | ||
| 653 | |a Numerical analysis | ||
| 653 | |a Linear equations | ||
| 653 | |a Boundary conditions | ||
| 653 | |a Optimization algorithms | ||
| 653 | |a Efficiency | ||
| 700 | 1 | |a Cao, Fujun |u School of Science, Inner Mongolia University of Science and Technology, Baotou 014010, China | |
| 700 | 1 | |a Ge, Yongbin |u School of Mathematical Statistics, Ningxia University, Yinchuan 750021, China; <email>371102a29nj.cdb@sin.cn</email>; School of Science, Dalian Minzu University, Dalian 116600, China | |
| 773 | 0 | |t Axioms |g vol. 14, no. 1 (2025), p. 75 | |
| 786 | 0 | |d ProQuest |t Engineering Database | |
| 856 | 4 | 1 | |3 Citation/Abstract |u https://www.proquest.com/docview/3159334398/abstract/embedded/7BTGNMKEMPT1V9Z2?source=fedsrch |
| 856 | 4 | 0 | |3 Full Text + Graphics |u https://www.proquest.com/docview/3159334398/fulltextwithgraphics/embedded/7BTGNMKEMPT1V9Z2?source=fedsrch |
| 856 | 4 | 0 | |3 Full Text - PDF |u https://www.proquest.com/docview/3159334398/fulltextPDF/embedded/7BTGNMKEMPT1V9Z2?source=fedsrch |