求解浅水波方程的并行物理信息神经网络算法

Concurrent PINN algorithm for solving shallow water wave equations

双曲守恒律方程是一类比较特殊的偏微分方程,其数值求解方法的研究一直是一个热点问题,一个显著特性是即使初始条件是光滑的,其解也可能会发展成间断。浅水波方程作为非线性双曲守恒律方程,由于间断解的存在,其精确求解存在很大困难。针对浅水波方程数值求解问题,本文基于PINN(Physics informed neural networks)反问题网络结构构造新的网络,构造的网络结构包括两个并行的神经网络,其中一个网络与已知状态数据(熵稳定格式加密求出)相关,另一个网络与方程本身相关。利用已知速度数据结合浅水波方程本身求解未知水深,最终通过一些数值算例验证网络的可行性。结果表明,新的网络结构可用于浅水波方程求解,利用速度数据可以较为精确地推算出水深。

Hyperbolic conservation equation is a special class of partial differential equations,and the study of its numerical solution method has always been a hot topic.One of its remarkable properties is that its solution may contain discontinuity even if the initial conditions are smooth.As a representation of the nonlinear hyperbolic conservation law,the shallow water wave equation is difficult to be solved precisely because of the existence of discontinuous solutions.In order to solve numerically the shallow water wave equation,a new network is constructed based on the inverse problem framework of PINN(Physics-informed Neural Networks).The network structure consists of two parallel neural networks,one of which is related to the known data obtained by the entropy stable schemes.The other network is related to the equation itself.The unknown water depth is determined by combining the known velocity data with the shallow water wave equation itself.Finally,the feasibility of the network is verified by some numerical examples.The results show that the new network structure can be used to solve the shallow water wave equation,and the water depth can be accurately calculated.