区域压缩PINN算法在双曲守恒律方程求解中的应用  

Application of region compression PINN algorithm in solving hyperbolic conservation laws

作　　者：张蕊 郑素佩[1] 董安国[1] 汪浏博 ZHANG Rui;ZHENG Su-pei;DONG An-guo;WANG Liu-bo(School of Science,Chang’an University,Xi’an 710064,China)

机构地区：[1]长安大学理学院,西安710064

出　　处：《计算力学学报》2024年第5期935-941,共7页Chinese Journal of Computational Mechanics

基　　金：国家自然科学基金(11971075,12101073);陕西省重点产业创新链项目(2020ZDLGY09-09)资助。

摘　　要：为捕捉双曲守恒律方程间断,提高算法求解精度,本文应用区域压缩PINN(physics-informed neural networks)算法对双曲守恒律方程近似求解。首先,对物理方程添加速度梯度监测函数,以此识别和压缩大梯度区域;随后,针对不同初始条件的双曲守恒律方程,设定相应的大梯度区域压缩控制系数,降低其在损失函数中占的比重;最后,将带有速度梯度权重项的损失函数放入神经网络中训练,通过最小化损失函数学习方程在整个区域上的解。利用区域压缩PINN算法求解各种经典双曲守恒律问题,通过对满足不同初始条件的一维和二维双曲守恒律方程进行数值模拟,并与经典PINN算法结果进行比较,验证了区域压缩PINN算法的良好性能。In order to capture the discontinuities and improve the accuracy of the algorithm,the region compression PINN algorithm is used to approximate the hyperbolic conservation laws.First of all,the velocity gradient monitoring function is added to the physical equation to identify and compress the large gradient region.Then,for hyperbolic conservation laws with different initial conditions,the corresponding compression control coefficient of the large gradient region is set to reduce its proportion in the loss function.Finally,the loss function with the velocity gradient weight term is put into the neural network for training,and through learning the solution of the equation over the entire region is obtained by minimizing the loss function.This paper applies this algorithm to various classical cases,by numerical simulation of one-dimensional and two-dimensional hyperbolic conservation laws satisfying different initial conditions.Compared with the results of classical PINN algorithm,the good performance of this algorithm is verified.

关 键 词：双曲守恒律方程 PINN算法 区域压缩 梯度权重 激波 

分 类 号：O241.82[理学—计算数学]