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融合傅里叶与自适应权重的物理信息神经网络波动方程求解

Solving Wave Equations with Physics-Informed Neural Networks Integrating Fourier Features and Adaptive Weights

  • 摘要: 针对传统的物理信息神经网络(physics-informed neural networks,PINN)在求解波动方程时难以有效捕捉高频信息,进而导致收敛速度缓慢、求解精度受限的问题,本文提出一种融合傅里叶特征映射与自适应权重机制的PINN改进模型(adaptive weight and Fourier feature PINN,AWF−PINN)。该模型首先在神经网络输入端引入可学习的傅里叶特征映射,以增强模型对高频振荡特征的表达能力;同时,基于各损失项的变化率设计指数归一化自适应权重策略,以动态平衡偏微分方程残差、初始条件及边界条件等多项损失项,从而提升优化稳定性。为验证模型性能,分别以一维波动方程和Klein−Gordon方程为测试案例进行数值实验。结果表明:与传统PINN相比,AWF−PINN在两类方程上均取得了显著的性能提升,不仅收敛所需迭代次数显著减少,而且相对误差降低达80%以上,有效缓解了频谱偏差与损失项失衡的问题。本文方法通过融合傅里叶特征嵌入与权重自适应机制,为求解具有周期性高频特征的偏微分方程提供了一种高效且鲁棒的求解途径,进一步拓展了PINN在复杂波动问题中的应用前景。

     

    Abstract: To mitigate the challenge that conventional physics-informed neural networks (PINN) face in effectively capture high-frequency information when solving wave equations, resulting in slow convergence and limited accuracy, an improved PINN model, termed adaptive weight and Fourier feature PINN (AWF-PINN), was proposed. In this approach, a learnable Fourier feature mapping was incorporated into the input layer to enhance the model’s capability for high-frequency oscillatory features. Additionally, an adaptive weight strategy based on exponential normalization of loss change rates was designed to dynamically balance multiple losses, including the PDE residual, initial conditions, and boundary conditions, thereby improving optimization stability. To validate the performance of the proposed model, numerical experiments were conducted on the wave equation and the Klein-Gordon equation as test cases.The results demonstrate that, compared with the traditional PINN, AWF-PINN achieves significant performance improvements on both equations, with substantially fewer iterations required for convergence and a reduction in relative errors by over 80%, thereby effectively mitigating the issues of spectral bias and loss term imbalance. By integrating Fourier feature embedding with weight adaptation, this study provides an efficient and robust approach for solving partial differential equations exhibiting periodic and high-frequency characteristics, thus broadening the applicability of PINNs to complex wave problems.

     

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