基于傅里叶特征映射的物理信息残差网络求解高阶非线性偏微分方程

doi:10.16088/j.issn.1001-6600.2026010602

Abstract

Abstract: An improved model named FF-ResPINN, combining Fourier feature mapping with residual connections, is proposed to address the challenges of spectral bias and gradient pathology encountered by Physics-Informed Neural Networks (PINN) when solving high-order nonlinear partial differential equations. Firstly, input coordinates are projected into a high-dimensional feature space by Fourier feature mapping. The spectral bandwidth of the Neural Tangent Kernel is regulated, and the capability of the network to capture high-frequency oscillatory solutionsis enhanced. Secondly, residual connections are introduced into the network to establish direct propagation paths for gradients. Consequently, the vanishing gradient problem triggered by high-order derivative calculations is effectively mitigated. The fourth-order Boussinesq equation and the fifth-order Sawada-Kotera equationare selected for numerical verification. Experimental results show that, compared with the standard PINN model, the solution accuracy and convergence stability are obviously improved by FF-ResPINN. The relative L₂ error reaches O(10^-5). The effectiveness of the proposed method in solving high-order nonlinear partial differential equations is verified.

Key words: physics-informed neural networks, Fourier feature mapping, residual network, high-order nonlinear partial differential equations, neural tangent kernel

CLC Number: O214.82

He Rongze, Wang Qingzhu, Xia Daoxun, Zhou Xiaojun. Physics-informed residual networks based on Fourier feature mapping for solving high-order nonlinear partial differential equations[J].Journal of Guangxi Normal University(Natural Science Edition), 2026, 44(6): 134-145.

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Physics-informed residual networks based on Fourier feature mapping for solving high-order nonlinear partial differential equations

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