Numerical solution of KdV-mKdV equation based on PINN and its improved method(基于PINN及其改进算法求解KdV-mKdV方程)

Abstract

Physics-informed neural network (PINN) had been proved to be an effective tool for solving partial differential equations and systems of equations. We used the PINN algorithm to obtain the numerical solution of the 1+1-dimensional KdV-mKdV equation, and the numerical results proved the reliability of the PINN algorithm. Although it has a high solution accuracy compared with the traditional methods, PINN's accuracy was overly dependent on the number of configuration points and weakness of large gradients. To overcome these defects, an improved PINN algorithm is proposed based on the idea of gradient enhancement, namely gradient-enhanced physics-informed neural network (gPINN) for solving the problem. gPINN algorithm makes up the defect of gradient attenuation by embedding the gradient information of partial differential equation residuals into the loss function. The results of numerical simulation of the 1+1-dimensional KdV-mKdV equation under different parameters show that the training error of the gPINN algorithm is reduced by one order of magnitude compared with the PINN algorithm when the number of configuration points is reduced by 2 orders of magnitude.(物理信息神经网络（physics-informed neural network，PINN）是求解偏微分方程及方程组的有效工具。数值结果证明了用PINN算法求解1+1维KdV-mKdV方程的可靠性，且求解精度较传统数值算法高，但求解精度过度依赖于训练点数，且易出现大梯度变弱的问题。为此，基于梯度增强思想提出了一种改进的PINN算法，即梯度增强物理信息神经网络（gradient-enhanced physics-informed neural network，gPINN）算法，通过将偏微分方程残差的梯度信息嵌入损失函数，弥补了梯度减弱的缺陷。用gPINN算法数值模拟了不同参数下1+1维KdV-mKdV方程，结果表明，gPINN算法在训练点数减少2个数量级的情况下，其训练误差仍比PINN算法减少一个数量级。)

Keywords

• pinn(物理信息神经网络)
• gpinn(梯度增强物理信息神经网络)
• 1+1-dimensional kdv-mkdv equation(1+1维kdv-mkdv方程)
• numerical simulation(数值模拟)