Transformed Diffusion-Wave fPINNs: Enhancing Computing Efficiency for PINNs Solving Time-Fractional Diffusion-Wave Equations
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| Format: | Preprint |
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2025
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| author | Li, Jing Zhang, Zhengqi |
| author_facet | Li, Jing Zhang, Zhengqi |
| contents | We propose transformed Diffsuion-Wave fractional Physics-Informed Neural Networks (tDWfPINNs) for efficiently solving time-fractional diffusion-wave equations with fractional order $α\in(1,2)$. Conventional numerical methods for these equations often compromise the mesh-free advantage of Physics-Informed Neural Networks (PINNs) or impose high computational costs when computing fractional derivatives. The proposed method avoids first-order derivative calculations at quadrature points by introducing an integrand transformation technique, significantly reducing computational costs associated with fractional derivative evaluation while preserving accuracy. We conduct a comprehensive comparative analysis applying this integrand transformation in conjunction with both Monte Carlo integration and Gauss-Jacobi quadrature schemes across various time-fractional PDEs. Our results demonstrate that tDWfPINNs achieve superior computational efficiency without sacrificing accuracy. Furthermore, we incorporate the proposed approach into adaptive sampling approaches such as the residual-based adaptive distribution (RAD) for the time-fractional Burgers equation with order $α\in(1,2)$, which exhibits complex solution dynamics. The experiments show that the Gauss-Jacobi method typically outperforms the Monte Carlo approach; however, careful consideration is required when selecting the number of quadrature points. Overall, the proposed tDWfPINNs offer a significant advancement in the numerical solution of time-fractional diffusion-wave equations, providing an accurate and scalable mesh-free alternative for challenging fractional models. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_11518 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Transformed Diffusion-Wave fPINNs: Enhancing Computing Efficiency for PINNs Solving Time-Fractional Diffusion-Wave Equations Li, Jing Zhang, Zhengqi Numerical Analysis We propose transformed Diffsuion-Wave fractional Physics-Informed Neural Networks (tDWfPINNs) for efficiently solving time-fractional diffusion-wave equations with fractional order $α\in(1,2)$. Conventional numerical methods for these equations often compromise the mesh-free advantage of Physics-Informed Neural Networks (PINNs) or impose high computational costs when computing fractional derivatives. The proposed method avoids first-order derivative calculations at quadrature points by introducing an integrand transformation technique, significantly reducing computational costs associated with fractional derivative evaluation while preserving accuracy. We conduct a comprehensive comparative analysis applying this integrand transformation in conjunction with both Monte Carlo integration and Gauss-Jacobi quadrature schemes across various time-fractional PDEs. Our results demonstrate that tDWfPINNs achieve superior computational efficiency without sacrificing accuracy. Furthermore, we incorporate the proposed approach into adaptive sampling approaches such as the residual-based adaptive distribution (RAD) for the time-fractional Burgers equation with order $α\in(1,2)$, which exhibits complex solution dynamics. The experiments show that the Gauss-Jacobi method typically outperforms the Monte Carlo approach; however, careful consideration is required when selecting the number of quadrature points. Overall, the proposed tDWfPINNs offer a significant advancement in the numerical solution of time-fractional diffusion-wave equations, providing an accurate and scalable mesh-free alternative for challenging fractional models. |
| title | Transformed Diffusion-Wave fPINNs: Enhancing Computing Efficiency for PINNs Solving Time-Fractional Diffusion-Wave Equations |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2506.11518 |