一种自适应加权外推神经网络用于KdV方程平方根初值条件产生的色散冲击波
An adaptive weighted extrapolation neural network for dispersive shock waves generated by square-root initial condition of the KdV equation
AI总结：
针对KdV方程平方根初值条件产生色散冲击波模拟中的数值困难，提出改进的物理引导多阶段神经网络IPgMSNN，通过在线微调和加权目标机制抑制外推误差累积，实现高精度参数反演与长期稳定模拟。
AI中文摘要：
使用物理信息神经网络（PINNs）模拟色散冲击波（DSWs）存在显著的数值困难。特别是，平方根初值条件在原点附近表现出奇异性，使得精确拟合变得困难。这种初始拟合误差通过非线性演化被放大，并在外推过程中逐渐累积，对数值方法的长期外推能力提出了很高的要求。为此，本文提出了一种改进的物理引导多阶段神经网络（IPgMSNN）框架。IPgMSNN建立在原始PgMSNN框架之上，其核心创新集中在第三阶段训练策略上。具体来说，引入了在线微调和加权目标机制，有效抑制了长期外推过程中的误差累积。数值实验从三个角度系统评估了IPgMSNN：正问题求解、模型稳定性和参数反演。实验结果表明，IPgMSNN能有效捕捉高频波前细节，保持长期外推稳定性，并实现高精度参数反演，为基于深度学习的色散非线性系统模拟提供了一种高效解决方案。
英文摘要：
Simulating dispersive shock waves (DSWs) using physics-informed neural networks (PINNs) presents significant numerical difficulties. In particular, the square-root initial condition exhibits singularity near the origin, making it difficult to fit accurately. This initial fitting error is amplified through nonlinear evolution and gradually accumulates during extrapolation, imposing high demands on the long-term extrapolation capability of numerical methods. Accordingly, this paper proposes an improved physics-guided multistage neural network (IPgMSNN) framework. IPgMSNN is built upon the original PgMSNN framework, with its core innovation focused on the third-stage training strategy. Specifically, online fine-tuning and a weighted target mechanism are introduced, which effectively suppress error accumulation during long-term extrapolation. Numerical experiments systematically evaluate IPgMSNN from three perspectives: forward problem solving, model stability, and parameter inversion. Experimental results demonstrate that IPgMSNN effectively captures high-frequency wavefront details, maintains long-term extrapolation stability, and achieves high-precision parameter inversion, providing an efficient solution for deep learning-based simulation of dispersive nonlinear systems.
