基于自适应权重两阶段PINNs方法求解偏微分方程

doi:10.16088/j.issn.1001-6600.2025050801

摘要/Abstract

摘要： 针对传统物理信息神经网络(PINNs)在处理高频特征时存在精度不足的问题,本文提出一种基于自适应权重两阶段PINNs方法(AWTS-PINNs)求解具有高频解的偏微分方程。该方法基于预训练和微调相结合的两阶段训练框架,引入具有高频特征响应能力的激活函数,并融合神经正切核自适应机制动态调节损失函数权重,从而显著提升模型对高频特征的表达与捕捉能力。实验结果表明,与PINNs、NTK-PINNs、RFF-PINNs和DG-PINNs等现有方法相比,AWTS-PINNs在捕捉高频特征方面表现出色,具有更高的精度和求解效率。在一维和二维数值实验中,AWTS-PINNs均取得最低测试误差,精度达到10^-4量级。

关键词: 物理信息神经网络, 高频解, 偏微分方程, 频谱偏差, 神经正切核

Abstract: Aiming at the problem of insufficient accuracy of traditional Physical information neural networks (PINNs) when dealing with high-frequency features, this paper proposes a two-stage PINNs method based on adaptive weights (AWTS-PINNs) to solve partial differential equations with high-frequency solutions. This method is based on a two-stage training framework combining pre-training and fine-tuning. It introduces an activation function with the response ability of high-frequency features and integrates the adaptive mechanism of neural tangent kernels to dynamically adjust the weight of the loss function, thereby significantly improving the model’s ability to express and capture high-frequency features. The experimental results show that this method performs well in capturing high-frequency features. Compared with the existing methods such as PINNs, NTK-PINNS, RFF-PINNS and DG-PINNs, AWTS-PINNs has higher accuracy and solution efficiency. In one-dimensional and two-dimensional numerical experiments, AWTS-PINNs achieves the lowest test errors, with an accuracy reaching the order of 10^-4.

Key words: PINNs, high-frequency solution, partial differential equations (PDEs), spectral bias, neural tangent kernel

中图分类号: O241.82

谢翔, 江林峰, 杨凤莲. 基于自适应权重两阶段PINNs方法求解偏微分方程[J]. 广西师范大学学报（自然科学版）, 2026, 44(2): 175-189.

XIE Xiang, JIANG Linfeng, YANG Fenglian. Two-Stage PINNs Method with Adaptive Weights for SolvingPartial Differential Equations[J]. Journal of Guangxi Normal University(Natural Science Edition), 2026, 44(2): 175-189.

参考文献

[1] 马亮, 马西奎, 迟明珺, 等.一种适用于嵌入式导电薄层的高阶电磁波混合时域有限差分-时程精细积分法[J].电工技术学报, 2025, 40(5):1333-1343.DOI:10.19595/j.cnki.1000-6753.tces.240355.
[2] 王雯宇, 程茜, 梁淇玮, 等.一维谐振子薛定谔方程含时演化非级数解析解[J].物理与工程, 2024, 34(4):98-109.DOI:10.3969/j.issn.1009-7104.2024.04.016.
[3] 魏健达, 张江敏.有限元方法求解二维薛定谔方程[J].福建师范大学学报(自然科学版), 2022, 38(1):24-33.DOI:10.12046/j.issn.1000-5277.2022.01.004.
[4] 高飞, 郭晓斌, 袁冬芳, 等.改进PINNs方法求解边界层对流占优扩散方程[J].广西师范大学学报(自然科学版), 2023, 41(6):33-50.DOI:10.16088/j.issn.1001-6600.2023032203.
[5] 韩烁, 江林峰, 杨建斌.基于注意力机制PINNs方法求解圣维南方程[J].广西师范大学学报(自然科学版), 2025, 43(4):58-68.DOI:10.16088/j.issn.1001-6600.2024061802.
[6] CAI X H, HUANG C J, TAO-RAN, et al.A mesh-free finite-difference scheme for frequency-domain acoustic wave simulation with topography[J].Applied Geophysics, 2023, 20(4):447-459.DOI:10.1007/s11770-022-0981-z.
[7] JIM DOUGLAS J.Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures[J].SIAM Journal on Numerical Analysis, 1982, 19(5):871-885.DOI:10.2307/2156980.
[8] ZHANG Y.A finite difference method for fractional partial differential equation[J].Applied Mathematics and Computation, 2009, 215(2):524-529.DOI:10.1016/j.amc.2009.05.018.
[9] TAYLOR C A, HUGHES T J R, ZARINS C K.Finite element modeling of blood flow in arteries[J].Computer Methods in Applied Mechanics and Engineering, 1998, 158(1/2):155-196.DOI:10.1016/S0045-7825(98)80008-X.
[10] QIN X Q, MA Y C, ZHANG Y.Two-grid method for characteristics finite-element solution of 2d nonlinear convection-dominated diffusion problem[J].Applied Mathematics and Mechanics, 2005, 26(11):1506-1514.DOI:10.1007/BF03246258.
[11] EYMARD R, GALLOUЁT T, HERBIN R.Finite volume methods[J].Handbook of Numerical Analysis, 2000, 7:713-1018.DOI:10.1016/S1570-8659(00)07005-8.
[12] RAISSI M, PERDIKARIS P, KARNIADAKIS G E.Physics-informed neural networks:a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations[J].Journal of Computational Physics, 2019, 378:686-707.DOI:10.1016/j.jcp.2018.10.045.
[13] MATTEY R, GHOSH S.A novel sequential method to train physics informed neural networks for Allen Cahn and Cahn Hilliard equations[J].Computer Methods in Applied Mechanics and Engineering, 2022, 390:114474.DOI:10.1016/j.cma.2021.114474.
[14] ZHOU W, XU Y F.Data-guided physics-informed neural networks for solving inverse problems in partial differential equations[EB/OL].(2024-07-15)[2025-05-08].https://doi.org/10.48550/arXiv.2407.10836.DOI:10.48550/arXiv.2407.10836.
[15] WANG S F, YU X L, PERDIKARIS P.When and why PINNs fail to train:a neural tangent kernel perspective[J].Journal of Computational Physics, 2022, 449:110768.DOI:10.1016/j.jcp.2021.110768.
[16] McCLENNY L D, BRAGA-NETO U M.Self-adaptive physics-informed neural networks[J].Journal of Computational Physics, 2023, 474:111722.DOI:10.1016/j.jcp.2022.111722.
[17] WANG S F, TENG Y J, PERDIKARIS P.Understanding and mitigating gradient flow pathologies in physics-informed neural networks[J].SIAM Journal on Scientific Computing, 2021, 43(5):A3055-A3081.DOI:10.1137/20m1318043.
[18] WANG S F, WANG H W, PERDIKARIS P.On the eigenvector bias of Fourier feature networks:From regression to solving multi-scale PDEs with physics-informed neural networks[J].Computer Methods in Applied Mechanics and Engineering, 2021, 384:113938.DOI:10.1016/j.cma.2021.113938.
[19] COOLEY M, SHANKAR V, KIRBY R M, et al.Fourier PINNs:from strong boundary conditions to adaptive Fourier bases[EB/OL].(2024-10-04)[2025-05-08].https://doi.org/10.48550/arXiv.2410.03496.DOI:10.48550/arXiv.2410.03496.
[20] RATHORE P, LEI W, FRANGELLA Z, et al.Challenges in training PINNs:a loss landscape perspective[EB/OL].(2024-06-03)[2025-05-08].https://doi.org/10.48550/arXiv.2402.01868.DOI:10.48550/arXiv.2402.01868.

Metrics

Viewed

Full text

Abstract

Cited

Shared

Discussed