王一然,卜涛,陈韵雅,吴子傲,张朋,蒋臣威,方爱平

• 1:西安交通大学物理学院

摘要(Abstract)：

本文研究了利用物理信息神经网络(Physics-informed Neural Network,简称PINN)求解偏微分方程中的本征值问题,并以二维定态薛定谔方程本征值问题为例进行了深入探讨。针对传统数值方法在复杂边界与高维问题求解中的局限性,本文提出了一种将本征能量E作为可训练参数,并结合Adam与L-BFGS混合优化策略的PINN求解框架。为了同时求解基态与多个激发态,在损失函数中引入了归一化约束和波函数间的正交性约束。研究选取了圆形、方形、五角星形及体育场形等不同几何边界进行测试。结果表明:PINN方法在规则及对称边界下,其低能级计算精度优于有限差分法;通过独立参数化与正交约束,能够有效识别因对称性导致的能级简并现象;在处理不规则边界时,其无网格特性避免了复杂网格生成的困难,展现了良好的适应性与潜力。凭借其无网格特性、对几何对称性的内禀保持能力以及求解高维问题的潜力,PINN为复杂边界下的本征值问题求解提供了一种有效且具有应用前景的新方法。

关键词(KeyWords)： 物理信息神经网络;本征值问题;偏微分方程;数值求解方法

Abstract:

Keywords:

基金项目(Foundation): 2025年度教育部高等学校大学物理课程教学指导委员会大中物理教育衔接工作委员会教学研究课题（WX202501）;; 教育部高等学校物理学类专业教学指导委员会高等学校热学课程教学研究会项目（JZW-25-RX-02）;; 西安交通大学本科教学改革研究项目（拔尖专项,24BJ01Z）;; 西安交通大学2026年课程思政专项研究项目

作者(Author): 王一然,卜涛,陈韵雅,吴子傲,张朋,蒋臣威,方爱平

DOI: 10.27024/j.wlygc.2026.03.17.02

参考文献(References)：

• [1] STRIKWERDA J C.Finite difference schemes and partial differential equations[M].Society for Industrial and Applied Mathematics,2004.
• [2] 陆金甫，关治.偏微分方程数值方法[M].2版.北京：清华大学出版社，2004.LU J F,GUAN Z.Numerical methods for partial differential equations[M].2nd ed.Beijing:Tsinghua University Press,2004.(in Chinese)
• [3] 王飞，党浩宁，尚勇，等.从传统数值方法到神经网络：偏微分方程数值解的演进与展望[J].现代应用物理，2025,16(1):41-66.WANG F,DANG H N,SHANG Y,et al.From Traditional Numerical Methods to Neural Networks:Evolution and Prospect of Numerical Solutions for Partial Differential Equations[J].Modern Applied Physics,2025,16(1):41-66.(in Chinese)
• [4] 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.
• [5] SARKAR S.Physics-informed neural networks for one-dimensional quantum well problems[J].arXiv:2504.05367,2025.
• [6] XU Z Q J,ZHANG Y,LUO T,et al.Frequency principle:Fourier analysis sheds light on deep neural networks[J].Communications in Computational Physics,2020,28(5):1746-1767.
• [7] 徐勃，李天驰，郭永锋，等.基于神经网络算法的菲涅耳衍射研究[J].物理与工程，2025,35(5):183-189,197.XU B,LI T C,GUO Y F,et al.Research on Fresnel diffraction based on neural network algorithms[J].Physics and Engineering,2025,35(5):183-189,197.(in Chinese)
• [8] 贾朔，车昊，李洋，等.基于BP神经网络的光纤陀螺误差补偿方法[J].物理与工程，2020,30(4):116-120.JIA S,CHE H,LI Y,et al.The fiber optic gyro error compensation by using BP neural network[J].Physics and Engineering,2020,30(4):116-120.(in Chinese)
• [9] RATHORE P,LEI W,FRANGELLA Z,LU L,& UDELL M.Challenges in Training PINNs:A Loss Landscape Perspective[J].arXiv.2402.01868,2024.
• [10] NOCEDAL J.Updating quasi-Newton matrices with limited storage[J].Mathematics of Computation,1980,35(151):773-782.
• [11] ZIENKIEWICZ O C,TAYLOR R L.The finite element method[M].Butterworth-Heinemann,2000.
• [12] REQUICHA A A G,& VOELCKER H B.Constructive solid geometry.Production Automation Project[M],University of Rochester,1977.
• [13] SMITH G D.Numerical solution of partial differential equations:finite difference methods[M].Oxford University Press,1985.
• [14] BERGSTRA J,BENGIO Y.Random search for hyper-parameter optimization[J].Journal of Machine Learning Research,2012,13:281-305.
• [15] ROBERT C P,CASELLA G.Monte Carlo statistical methods[M].Springer Science & Business Media,2004.
• [16] WANG S,TENG Y,PERDIKARIS P.Understanding and mitigating gradient pathologies in physics-informed neural networks[J].SIAM Journal on Scientific Computing,2021,43(5),A3055-A3081.
• [17] LIU M,YU S,LI Z,et al.Physics-informed neural network with fourier features for acoustic scattering by underwater targets[J].Journal of Physics:Conference Series,2026,3170:012015.
• [18] CHEN J,SHI J,HE A,et al.Data-driven solutions and parameter estimations of a family of higher-order KdV equations based on physics informed neural networks[J].Scientific Reports,2024,14:23874.
• [19] YU J,LU L,MENG X,et al.Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems[J].Computer Methods in Applied Mechanics and Engineering,2022,393:114823.