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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2026, Vol. 43 ›› Issue (2): 345-357.doi: 10.3969/j.issn.1005-3085.2026.02.010cstr: 32411.14.cjem.CN61-1269/O1.2026.02.010

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求解等离子体中的Schamel-Zakharov-Kuznetsov-Burgers's方程的深度学习方法

徐彩艳,  郭士民   

  1. 西安交通大学数学与统计学院，西安 710049
  • 收稿日期:2023-07-24 接受日期:2023-11-14 出版日期:2026-04-15 发布日期:2026-06-15

Solving the Schamel-Zakharov-Kuznetsov-Burgers Equation in Plasma Based on Deep Learning Method

XU Caiyan,  GUO Shimin   

  1. School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049
  • Received:2023-07-24 Accepted:2023-11-14 Online:2026-04-15 Published:2026-06-15

摘要：

为研究磁化离子对等离子体中的非线性波动现象，建立了三维Schamel-Zakharov-Kuznetsov-Burgers's方程。采用物理信息神经网络(Physics-Informed Neural Networks, PINN)对该非线性微分方程进行求解。针对PINN中由梯度流刚性引起的梯度不平衡问题及其固有的“谱偏移”现象，引入了自适应权重系数和傅里叶映射来改进算法。基于改进后的PINN模型，系统研究了粘性系数对激波特性的影响。数值结果表明，粘性系数的增大将导致激波趋于平滑。

关键词: 离子对等离子体, 物理信息神经网络, 神经正切核, 谱偏移

Abstract:

To investigate the nonlinear wave phenomena in magnetized ionic-pair plasma, a $(3+1)$-dimensional Schamel-Zakharov-Kuznetsov-Burgers' equation is established. The physics-informed neural network (PINN) is employed to solve this nonlinear differential equation. In order to address the unbalanced gradients caused by the stiffness of gradient flow dynamics and the inherent spectral bias in PINN, an adaptive weighting coefficient and Fourier feature mapping are introduced to improve the algorithm. By utilizing the modified PINN model, the influence of the viscosity coefficient on shock wave characteristics is systematically studied. Numerical results indicate that an increase in the viscosity coefficient leads to a smoothing effect on the shock wave structure.

Key words: magnetized ionic-pair plasma, PINN, neural tangent kernel, spectral bias

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