浅水波方程的黏性正则化PINN算法

Viscous Regularization PINN Algorithm for Shallow Water Equations

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。

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 onedimensional 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.