物理信息神经网络求解五阶emKdV方程的正反问题

Solving the Forward and Inverse Problems of Extended Fifth-Order mKdV Equation Via Physics-Informed Neural Networks

该文利用物理信息神经网络(PINNs)对扩展的五阶mKdV(emKdV)方程的正反问题进行求解,并对孤子的动力学行为进行分析、模拟.针对正问题,选用双曲正切函数tanh作为激活函数求解方程的一、二、三孤子解,并将PINNs方法求得的数据驱动解与借助简化的Hirota方法给出的方程精确解进行比较,一孤子解的精度为O(10^(-4)),二、三孤子解的精度为O(10^(-3)).针对反问题,分别由一、二、三孤子解的数据进行驱动求解方程的两个待定系数,并在不同的噪声下探究算法的鲁棒性.当在训练数据中加入1%的初始噪声或观测噪声时,待求系数的预测精度可分别达到O(10^(-3))和O(10^(-2));当加入3%的初始噪声或观测噪声时,预测精度依然可以达到O(10^(-2));由实验数据分析可知观测噪声对PINNs模型的影响要略大于初始噪声.

With the help of the physics-informed neural networks(PINNs),the forward and inverse problems of extended fifth-order mKdV(emKdV)equation are tackled,and the dynamic behaviors of solitons are also analyzed and simulated in this paper.The hyperbolic tangent function tanh is selected as the activation function to solve the one,two and three-soliton solutions of the equation.Moreover,the data-driven solutions obtained by PINNs method are compared with the exact solution given by the simplified Hirota method.Specifically,the accuracy of one-soliton solution is O(10^(-4)),and the accuracy of the two-soliton and three-soliton solutions is O(10^(-3)).For the inverse problem,the coeficients of the equation are discovered by the data of one,two and three-soliton solutions,respectively.Meanwhile,the robustness of the PINNs algorithm is explored under diferent noises.The accuracy of the datadriven coeficients can reach O(10^(-3))or O(10^(-2))respectively,when 1%initial noise or observation noise is added to the training data.And the prediction accuracy can stil reach O(10^(-2))even if 3%initial noise or observation noise is added.According to the analysis of experimental data,the impact of observation noise on PINNs model is slightly greater than the initial noise.