| Citation: |
Yanting Li, Huantian Xie, Juan Zhang, Jianwei Zhou. A CONCISE TRAINING SCHEME OF RBF NEURAL NETWORKS WITH FIXED CENTER POINTS FOR PARTIAL DIFFERENTIAL EQUATIONS WITH DIRICHLET BOUNDARY CONDITIONS[J]. Journal of Applied Analysis & Computation, 2026, 16(4): 1805-1821. doi: 10.11948/20250078
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A CONCISE TRAINING SCHEME OF RBF NEURAL NETWORKS WITH FIXED CENTER POINTS FOR PARTIAL DIFFERENTIAL EQUATIONS WITH DIRICHLET BOUNDARY CONDITIONS
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Abstract
A concise numerical scheme based on radial basis function neural networks is proposed for solving second order partial differential equations with Dirichlet boundary conditions. By employing fixed center points across the geometric region, our approach reduces the memory requirements. And the shape parameters are obtained by machine learning, which overcomes the shortcomings of artificially selection about parameters. Specially, the corresponding numerical schemes are calculated by nonlinear least squares problems, avoiding directly solving linear algebraic equations. The proposed training scheme of RBF neural networks, which bases on Gaussian and multiquadric radial basis functions, significantly achieves high-accuracy approximations and improves the computational efficiency compared to existing meshless methods. In comparison with Kansa's method and the radial basis function neural networks method proposed in [22], numerical experiments demonstrate that our approximation schemes perform well in various domains. Through comprehensive numerical experiments comparing Kansa's method and radial basis function neural networks, our proposed approximation schemes also exhibit better performance across multiple geometric domains, such as unit square, peanut-shaped and five-petaled domains.
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References
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Yanting Li, Huantian Xie, Juan Zhang, Jianwei Zhou. A CONCISE TRAINING SCHEME OF RBF NEURAL NETWORKS WITH FIXED CENTER POINTS FOR PARTIAL DIFFERENTIAL EQUATIONS WITH DIRICHLET BOUNDARY CONDITIONS[J]. Journal of Applied Analysis & Computation, 2026, 16(4): 1805-1821. doi: 10.11948/20250078
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Figure 1.
Architecture of RBF neural networks.
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Figure 2.
Centers(red) and collocation points (blue) of square.
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Figure 3.
Point-wise errors on the square with Nint = 16 and λ = 100.
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Figure 4.
Convergence curves on the square.
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Figure 5.
Centers(red) and collocation points (blue) of the peanut-shaped domain.
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Figure 6.
Point-wise errors on peanut-shaped domain with Nint = 20 and λ = 100.
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Figure 7.
Convergence curves on peanut-shaped domain.
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Figure 8.
Centers(red) and collocation points (blue) of five-petaled region.
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Figure 9.
Point-wise errors on five-petaled domain with
$ N_\text{int}=40 $ $ \lambda=100 $ -
Figure 10.
Convergence curves on five-petaled domain.
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Figure 11.
Convergence curves of three schemes.