机器学习在求解一维双曲守恒律方程中的应用

Application of machine learning in solving one-dimensional hyperbolic conservation law equation

双曲守恒律方程对空气动力学、物理学和海洋学等众多领域问题的计算有着重大意义,本文应用机器学习框架下的BP神经网络对双曲守恒律方程近似求解。首先,采用熵稳定格式及基于自适应移动网格的熵稳定格式所得多个时间层的数值解构造网络输入,采用高分辨率熵稳定格式所得对应的多个时间层的数值解构造网络输出,并对数据集作归一化处理。随后,利用三层的BP神经网络训练数据,从而得到性能良好的神经网络,以实现对任一给定时间层节点处数值解的预测。最后,通过五个数值算例表明该算法适用于该类问题的解决,数值结果分辨率高,且无非物理振荡产生。

The hyperbolic conservation equation is of great significance in the calculation of aerodynamics, physics, oceanography and many other fields.In this paper, the BP neural network based on a machine learning framework is applied to solve the hyperbolic conservation equation approximately.First of all, the algorithm constructs the network input from the numerical solutions of multiple time layers obtained from the entropy stable scheme and based on the adaptive moving grid, and uses the numerical solution of the corresponding multiple time layers obtained by the high-resolution entropy stable scheme to construct the network output, and the data set is normalized.Then, using the training data of three-layer BP neural network, the neural network with good performance is obtained, so as to realize the prediction of the numerical solutions at any given time level node.Finally, five numerical examples show that the algorithm is suitable for solving this kind of problems, and it has high resolution with no physical oscillations.