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间断问题扩散正则化的PINN反问题求解算法

林云云 郑素佩 封建湖 靳放

林云云,郑素佩,封建湖,靳放. 间断问题扩散正则化的PINN反问题求解算法 [J]. 应用数学和力学,2023,44(1):112-122 doi: 10.21656/1000-0887.430010
引用本文: 林云云,郑素佩,封建湖,靳放. 间断问题扩散正则化的PINN反问题求解算法 [J]. 应用数学和力学,2023,44(1):112-122 doi: 10.21656/1000-0887.430010
LIN Yunyun, ZHENG Supei, FENG Jianhu, JIN Fang. Diffusive Regularization Inverse PINN Solutions to Discontinuous Problems[J]. Applied Mathematics and Mechanics, 2023, 44(1): 112-122. doi: 10.21656/1000-0887.430010
Citation: LIN Yunyun, ZHENG Supei, FENG Jianhu, JIN Fang. Diffusive Regularization Inverse PINN Solutions to Discontinuous Problems[J]. Applied Mathematics and Mechanics, 2023, 44(1): 112-122. doi: 10.21656/1000-0887.430010

间断问题扩散正则化的PINN反问题求解算法

doi: 10.21656/1000-0887.430010
基金项目: 国家自然科学基金(11971075);陕西省自然科学基金青年项目(2020JQ-338;2020JQ-342)
详细信息
    作者简介:

    林云云(1994—),女,硕士生(E-mail:1121507157@qq.com

    郑素佩(1978—),女,副教授,博士,硕士生导师 (通讯作者. E-mail:zsp2008@chd.edu.cn)

  • 中图分类号: O241.82; O354.5

Diffusive Regularization Inverse PINN Solutions to Discontinuous Problems

  • 摘要:

    双曲守恒律方程间断问题的求解是该类方程数值求解问题研究的重点之一。 采用PINN (physics-informed neural networks)求解双曲守恒律方程正问题时需要添加扩散项,但扩散项的系数很难确定,需要通过试算方法来得到,造成很大的计算浪费。 为了捕捉间断并节约计算成本,对方程进行了扩散正则化处理,将正则化方程纳入损失函数中,使用守恒律方程的精确解或参考解作为训练集,学习出扩散系数,进而预测出不同时刻的解。 该算法与PINN求解正问题方法相比,间断解的分辨率得到了提高,且避免了多次试算系数的麻烦。 最后,通过一维和二维数值试验验证了算法的可行性,数值结果表明新算法捕捉间断能力更强、无伪振荡和抹平现象的产生,且所学习出的扩散系数为传统数值求解格式构造提供了依据。

  • 图  1  网络结构示意图

    Figure  1.  The diagram of the network structure

    图  2  例1的数值结果对比图:(a) 未正则化方程的精确解与预测解;(b) 正则化方程的精确解与预测解

    Figure  2.  Comparison of numerical results for example 1: (a) exact and predicted solutions to unregularized equations; (b) exact and predicted solutions to regularized equations

    图  3  例2的数值结果对比图:(a) 未正则化方程的精确解与预测解;(b) 正则化方程的精确解与预测解

    Figure  3.  Comparison of numerical results for example 2: (a) exact and predicted solutions to unregularized equations; (b) exact and predicted solutions to regularized equations

    图  4  例3的数值结果对比图:(a) 未正则化方程的精确解与预测解;(b) 正则化方程的精确解与预测解

    Figure  4.  Comparison of numerical results for example 3: (a) exact and predicted solutions to unregularized equations; (b) exact and predicted solutions to regularized equations

    图  5  例4的数值结果对比图:(a) 未正则化方程的精确解与预测解;(b) 正则化方程的精确解与预测解

    Figure  5.  Comparison of numerical results for example 4: (a) exact and predicted solutions to unregularized equations; (b) exact and predicted solutions to regularized equations

    图  6  例5的数值结果对比图:(a) 未正则化方程的精确解与预测解;(b) 正则化方程的精确解与预测解

    Figure  6.  Comparison of numerical results for example 5: (a) exact and predicted solutions to unregularized equations; (b) exact and predicted solutions to regularized equations

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  • 被引次数: 0
出版历程
  • 收稿日期:  2022-01-14
  • 录用日期:  2022-04-27
  • 修回日期:  2022-03-17
  • 网络出版日期:  2022-12-02
  • 刊出日期:  2023-01-15

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