双曲型方程激波捕捉的物理信息神经网络(PINN)算法  

PINN-type algorithm for shock capturing of hyperbolic equations

作　　者：郑素佩[1] 靳放 封建湖[1] 林云云 ZHENG Supei;JIN Fang;FENG Jianhu;LIN Yunyun(School of Science,Chang′an University,Xi′an 710064,China)

机构地区：[1]长安大学理学院,陕西西安710064

出　　处：《浙江大学学报（理学版）》2023年第1期56-62,82,共8页Journal of Zhejiang University（Science Edition）

基　　金：国家自然科学基金资助项目(11971075);陕西省自然科学基金青年项目(2020JQ-338,2020JQ-342)。

摘　　要：双曲型方程的数值求解算法研究一直是偏微分方程研究的热点,其中,双曲型方程的间断捕捉是难点。受物理信息神经网络(physics-informed neural networks,PINN)启发,构造了改进的PINN算法,近似求解双曲型方程的间断问题。将坐标构造的数据集作为神经网络的输入,将PINN算法中的损失函数作为训练输出值与参考解(基于细网格的熵相容格式数据)或准确解的误差值,通过网络优化,最小化损失函数,得到最优网络参数。最后用数值算例验证了算法的可行性,数值结果表明,本文算法能捕捉激波,分辨率高,且未产生伪振荡。The numerical solution of hyperbolic equation is a well-know hot topic in the field of numerical solution of partial differential equation,among which the discontinuous capturing of hyperbolic equation is always a difficult problem.Inspired by physical-informed neural networks(PINN),this paper presents a PINN-type algorithm to approximately solve discontinuity problem of hyperbolic equations.It takes the data set constructed by coordinate as the input of neural network.The loss function in PINN algorithm is converted to the error between the output value of the training network and the reference solution(entropy compatible format data based on the fine grid)or the exact solution.Then the loss function is minimized by network optimization to obtain the optimal network parameters.Finally,some numerical examples are demonstrated to verify the feasibility of the proposed algorithm.The numerical results show that the proposed algorithm can capture shock waves,and it has high resolution,without nonphysical oscillations.

关 键 词：双曲守恒律方程 网络预测 物理信息神经网络(PINN) 激波捕捉 

分 类 号：O241.8[理学—计算数学]