浅水波方程的黏性正则化PINN算法

doi:10.19596/j.cnki.1001-246x.8592

计算物理 ›› 2023, Vol. 40 ›› Issue (3): 314-324.DOI: 10.19596/j.cnki.1001-246x.8592

浅水波方程的黏性正则化PINN算法

郑素佩(), 林云云*(), 封建湖, 靳放

长安大学理学院, 陕西西安 710064

收稿日期:2022-07-15 出版日期:2023-05-25 发布日期:2023-07-22
通讯作者: 林云云
作者简介:
郑素佩(1978-)，女，博士，副教授，研究方向为科学与工程中的高性能计算技术，E-mail：zsp2008@chd.edu.cn
基金资助:
国家自然科学基金(11971075); 国家自然科学基金(11901057)

Viscous Regularization PINN Algorithm for Shallow Water Equations

Supei ZHENG(), Yunyun LIN*(), Jianhu FENG, Fang JIN

School of Science, Chang'an University, Xi'an, Shaanxi 710064, China

Received:2022-07-15 Online:2023-05-25 Published:2023-07-22
Contact: Yunyun LIN

摘要/Abstract

摘要：

针对经典PINN(Physics-informed Neural Networks)在求解浅水波方程间断问题时的不足，提出一种黏性耗散机制的正则化PINN算法。该算法利用黏性正则化的浅水波方程作为网络构建中的物理约束，并在损失函数中作为惩罚项，训练网络用正则化方程的光滑解逼近原方程的间断解，采用网格加密熵稳定格式的数值解作为参考，学习得原方程在整个区域的解。对满足不同初始条件的一维、二维浅水问题进行数值模拟，并与经典PINN算法进行比较，数值结果表明新算法泛化能力强，可预测任意时刻的解，分辨率高，不会出现抹平和伪振荡现象。

关键词: 浅水波方程, PINN算法, 黏性正则化, 黏性消失解

Abstract:

Because of the shortcomings of classical PINN (Physical-informed Neural Networks) for discontinuous problems of shallow water equation, a regularized PINN algorithm based on viscous dissipative mechanism was proposed. In the network framework, the viscous regularized shallow water equation is used as the physical constraint and the penalty term in the loss function. Training network makes the smooth solution of the regularized equation approximate the discontinuous solution of the original equation. Finally, for one-dimensional and two-dimensional shallow water problems with different initial conditions, the numerical results show that the new algorithm has strong generalization ability, can predict the solution at any time, and has high resolution, without the phenomenon of spurious oscillation.

Key words: shallow water equation, PINN algorithm, viscous regularization, viscous vanishing solution

郑素佩, 林云云, 封建湖, 靳放. 浅水波方程的黏性正则化PINN算法[J]. 计算物理, 2023, 40(3): 314-324.

Supei ZHENG, Yunyun LIN, Jianhu FENG, Fang JIN. Viscous Regularization PINN Algorithm for Shallow Water Equations[J]. Chinese Journal of Computational Physics, 2023, 40(3): 314-324.

图/表 7

图1 PINN的结构示意图

Fig.1 Diagrammatic sketch of PINN

图2 粘性正则化PINN的结构示意图

Fig.2 Diagrammatic sketch of viscous regularization PINN

图3 算例1结果(a)经典PINN算法的结果；(b)正则化PINN算法的结果

Fig.3 Results on Exp. 1 (a)results of the classical PINN algorithm; (b)results of the regularization PINN algorithm

图4 算例2结果(a)经典PINN算法的结果；(b)正则化PINN算法的结果

Fig.4 Results on Exp. 2 (a)results of the classical PINN algorithm; (b)results of the regularization PINN algorithm

图5 算例3结果(a)经典PINN算法的结果；(b)正则化PINN算法的结果

Fig.5 Results on Exp. 3(a)results of the classical PINN algorithm; (b)results of the regularization PINN algorithm

图6 算例4结果(t=0.2) (a)经典PINN算法的结果；(b)正则化PINN算法的结果

Fig.6 Results on Exp. 4(t=0.2) (a)results of the classical PINN algorithm; (b)results of the regularization PINN algorithm

图7 算例5结果(t=0.45) (a)经典PINN算法的结果；(b)正则化PINN算法的结果

Fig.7 Results on Exp. 5 (t=0.45) (a)results of the classical PINN algorithm; (b)results of the regularization PINN algorithm

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浅水波方程的黏性正则化PINN算法

Viscous Regularization PINN Algorithm for Shallow Water Equations

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