微分方程特征值问题的物理信息神经网络数值解法

doi:10.6040/j.issn.1671-9352.0.2024.244

摘要/Abstract

摘要： 针对微分方程特征值问题,提出一个改进的物理信息神经网络求解方法和两阶段训练法。该方法可以求解多个绝对值最小特征值、求解离初始值最近的特征值问题和重特征值问题等。通过一维、二维正方形区域以及L形区域中拉普拉斯算子特征值问题的数值算例表明本文方法比现有方法精度更高。

关键词: 物理信息神经网络, 微分方程, 特征值问题

Abstract: For eigenvalue problems of differential equations, an improved physical information neural network solution and a two-stage training method are proposed. The new method can solve multiple minimum eigenvalues, the eigenvalue problem closest to the initial value and the multiple eigenvalue problem. Numerical examples of Laplace operator eigenvalue problems in 1D and 2D square regions as well as L-shaped regions show that the new method is more accurate than the existing methods.

Key words: physically informed neural networks, differential equations, eigenvalue problems

中图分类号:

O242

唐瑜,袁利军Symbol`@@. 微分方程特征值问题的物理信息神经网络数值解法[J]. 《山东大学学报(理学版)》, 2026, 61(2): 26-36.

TANG Yu, YUAN Lijun. Numerical solution of physically informed neural networks for eigenvalue problems of differential equations[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2026, 61(2): 26-36.

参考文献

[1] 李蕾,叶永升. 具有Dirichlet有界条件的反应扩散Cohen-Grossberg神经网络指数稳定性[J]. 山东大学学报(理学版),2023,58(10):67-74. LI Lei, YE Yongsheng. Exponential stability of reaction-diffusion Cohen-Grossberg neural networks with Dirichlet boundary conditions[J]. Journal of Shandong University(Natural Science), 2023, 58(10):67-74.
[2] 王新生,朱小飞,李程鸿. 标签指导的多尺度图神经网络蛋白质作用关系预测方法[J]. 山东大学学报(理学版),2023,58(12):22-30. WANG Xinsheng, ZHU Xiaofei, LI Chenghong. Label guided multi-scale graph neural network for protein-protein interaction prediction[J]. Journal of Shandong University(Natural Science), 2023, 58(12):22-30.
[3] 徐光柱,刘鸣,任东,等. 基于脉冲耦合神经网络的多区域图像分割[J]. 山东大学学报(理学版),2010, 45(7):86-93. XU Guangzhu, LIU Ming, REN Dong, et al. Multi-region image segmentation based on a pulse coupled neural network[J]. Journal of Shandong University(Natural Science), 2010, 45(7):86-93.
[4] HUANG S D, FENG W T, TANG C W, et al. Partial differential equations meet deep neural networks: a survey [EB/OL].(2022-03-17)[2025-03-18]. https://arxiv.org/abs/2211.05567v2.
[5] E W N. The dawning of a new era in applied mathematics[J]. Notices of the American Mathematical Society, 2021, 68(4):565-571
[6] HAN J Q, JENTZEN A, E W N. Solving high-dimensional partial differential equations using deep learning[J]. Proceedings of the National Academy of Science, 2018, 115(34):8505-8510.
[7] E W N, YU B. the deep Ritz method:a deep learning-based numerical algorithm for solving variational problems[J]. Communications in Mathematics and Statistics, 2018, 6(1):1-12.
[8] SIRIGNANO J, SPILIOPOULOS K. DGM: a deep learning algorithm for solving partial differential equations[J]. Journal of Computational Physics, 2018, 375:1339-1364.
[9] 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.
[10] ZANG Y H, BAO G, YE X J, et al. Weak adversarial networks for high-dimensional partial differential equations[J]. Journal of Computational Physics, 2020, 411:109409.
[11] HAO Z K, LIU S M, ZHANG Y C, et al. Physics-informed machine learning: a survey on problems, methods and applications[EB/OL].(2023-03-07)[2025-03-18]. https://arxiv.org/abs/2211.08064.
[12] CAI S Z, MAO Z P, WANG Z C, et al. Physics-informed neural networks(PINNs)for fluid mechanics: a review[J]. Acta Mechanica Sinica, 2021, 37(12):1727-1738.
[13] CUOMO S, DICOLA V S, GIAMPAOLO F, et al. Scientific machine learning through physics-informed neural networks: where we are and whats next[J]. Journal of Scientific Computing, 2022, 92(3):88.
[14] RATHORE P, LEI W M, FRANGELLA Z, et al. Challenges in training PINNs: a loss landscape perspective[EB/OL].(2024-06-03)[2025-03-18]. https://arxiv.org/abs/2402.01868.
[15] ANAGNOSTOPOULOS S J, TOSCANO J D, STERGIOPULOS N, et al. Learning in PINNs: phase transition, total diffusion, and generalization[EB/OL].(2024-03-27)[2025-03-18]. https://arxiv.org/abs/2403.18494.
[16] LU L, MENG X H, MAO Z P, et al. DeepXDE: a deep learning library for solving differential equations[J]. SIAM Review, 2021, 63(1):208-228.
[17] LU L, JIN P Z, PANG G F, et al. Learning nonlinear operators via Deep ONet based on the universal approximation theorem of operators[J]. Nature Machine Intelligence, 2021, 3:218-229.
[18] CAO Q Y, GOSWAMI S, KARNIADAKIS G E. Laplace neural operator for solving differential equations[J]. Nature Machine Intelligence, 2024, 6:631-640.
[19] HAN J Q, LU J F, ZHOU M. Solving high-dimensional eigenvalue problems using deep neural networks: a diffusion Monte Carlo like approach[J]. Journal of Computational Physics, 2020, 423:109792.
[20] YANG Q H, DENG Y T, YANG Y, et al. Neural networks based on power method and inverse power method for solving linear eigenvalue problems[J]. Computers & Mathematics with Applications, 2023, 147:14-24.
[21] BEN-SHAUL I, BAR L, FISHELOV D, et al. Deep learning solution of the eigenvalue problem for differential operators[J]. Neural Computation, 2023, 35(6):1100-1134.
[22] LIU Z Q, CAI W, XU Z J. Multi-scale deep neural network(MscaleDNN)for solving Poisson-Boltzmann equation in complex domains[J]. Communications in Computational Physics, 2020, 28(5):1970-2001.

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