一类分数阶扩散方程的反问题
——基于谱方法和PINNs方法
PDF下载 (85)邬倩文,马飞遥.
一类分数阶扩散方程的反问题
——基于谱方法和PINNs方法
[J].宁波大学学报(理工版),2025,38(6):91-96.DOI:10.20098/j.cnki.1001-5132.2025.0314WU Qianwen,MA Feiyao.An inverse problem of fractional diffusion equations: based on the spectral method and PINNs method[J].Journal of Ningbo University(Natural Science & Engineering Edition),2025,38(6):91-96.DOI:10.20098/j.cnki.1001-5132.2025.0314
| Title: | An inverse problem of fractional diffusion equations: based on the spectral method and PINNs method |
| 作者: | 邬倩文, 马飞遥 |
| Author(s): | WU Qianwen, MA Feiyao |
| 关键词: | 分数阶扩散方程; 参数反演; 物理信息神经网络; 延拓法; 谱方法 |
| Keywords: | fractional diffusion equations; parameter inversion; physics-informedneural networks; extension technique; spectral method |
| 分类号: | O241.82；O175.2 |
| DOI: | 10.20098/j.cnki.1001-5132.2025.0314 |
| 文献标识码: | A |
| 摘要: | 提出了一种基于物理信息神经网络（Physics-Informed Neural Networks，PINNs）的方法，用于求解无界区域分数阶扩散方程的参数反演问题。首先用延拓法将非局部的分数阶扩散方程转换为高一维局部方程，克服非局部数值求解的困难；然后用PINNs方法解出局部方程谱的展开系数和反演参数值，解决了无界区域数值求解问题；最后通过数值实验验证该方法的有效性。 |
| Abstract: | This paper proposes an approach based on Physics-Informed Neural Networks (PINNs) to address the parameter inversion problem for fractional diffusion equations on unbounded domains, which lack effective algorithms. Firstly, the non-local fractional diffusion equation is transformed into a higher-dimensional local equation through the extension technique to overcome numerical difficulties caused by nonlocality. Subsequently, PINNs are utilized to determine both the spectral expansion coefficients of the local equation and the inversion parameters values, addressing numerical challenges on unbounded domains. Lastly, the effectiveness of the proposed algorithm is demonstrated with numerical experiments. |
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| 备注/Memo: | 收稿日期：2025−03−20 宁波大学学报（理工版）网址：http://journallg.nbu.edu.cn/ 基金项目：国家自然科学基金（11971251） 第一作者：邬倩文，硕士研究生，主要研究方向为偏微分方程。E-mail: 378920756@qq.com *通信作者：马飞遥，博士/副教授，主要研究方向为偏微分方程。E-mail: mafeiyao@nbu.edu.cn 宁波大学学报（理工版）网址：http://journallg.nbu.edu.cn/ |
