广西师范大学学报（自然科学版） ›› 2023, Vol. 41 ›› Issue (6): 33-50.doi: 10.16088/j.issn.1001-6600.2023032203

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改进PINNs方法求解边界层对流占优扩散方程

高飞1, 郭晓斌1, 袁冬芳2, 曹富军2*   

  1. 1.内蒙古科技大学 信息工程学院, 内蒙古 包头 014010;
    2.内蒙古科技大学 理学院, 内蒙古 包头 014010
  • 收稿日期:2023-03-22 修回日期:2023-04-24 发布日期:2023-12-04
  • 通讯作者: 曹富军(1984—), 男, 宁夏中卫人, 内蒙古科技大学副教授, 博士。 E-mail: caofujun@imust.edu.cn
  • 基金资助:
    国家自然科学基金(12161067, 12261067); 内蒙古自然科学基金(2021LHMS01006, 2022MS01008); 内蒙古自治区青年科技英才支持计划项目(NJYT20B15); 内蒙古科技大学创新基金(2019YQL02)

Improved PINNs Method for Solving the Convective Dominant Diffusion Equation with Boundary Layer

GAO Fei1, GUO Xiaobin1, YUAN Dongfang2, CAO Fujun2*   

  1. 1. School of Information Engineering, Inner Mongolia University of Science and Technology, Baotou Neimenggu 014010, China;
    2. School of Science, Inner Mongolia University of Science and Technology, Baotou Neimenggu 014010, China
  • Received:2023-03-22 Revised:2023-04-24 Published:2023-12-04

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摘要： 针对物理信息神经网络(PINNs)在求解边界层附近存在剧烈梯度变化的对流占优扩散方程时无法得到足够精度的问题,本文提出一种具有参数渐进思想的神经网络求解方法。该方法首先近似大扩散参数方程的光滑解,然后逐步减小扩散参数并将大扩散参数下的网络最优参数作为小扩散参数神经网络的初始值进行训练,通过参数循环反复优化物理信息神经网络,提高神经网络的表征能力,从而提升物理信息神经网络逼近对流占优扩散问题的求解精度,最后获得小扩散参数的高精度奇异解。经过对本文方法与PINNs以及gPINNs方法在精度和收敛效率方面的对比分析表明,本文方法在未知边界层位置条件下,能够高效地近似对流占优扩散方程的大梯度解,实现10-3量级的精度。同时,本文方法在收敛速度和稳定性方面比PINNs和gPINNs具有更好的优势和性能。

关键词: 扩散方程, 边界层, 物理信息神经网络, 深度学习, 对流扩散

Abstract: To address the issue of insufficient accuracy when solving convection-dominated diffusion equations with drastic gradient changes near the boundary layer using Physics-Informed Neural Networks (PINNs), a neural network solution method incorporating a parameter progressive strategy is proposed. This method initially approximates the smooth solution of the large diffusion parameter equation, then progressively reduces the diffusion parameter while using the optimal network parameters from the large diffusion parameter as the initial values for training the small diffusion parameter neural network. By iteratively optimizing the Physics-Informed Neural Networks through parameter cycling, the neural network’s representational capacity is enhanced, thereby improving the approximation accuracy of the convection-dominated diffusion problem. And finally a high-precision singular solution for the small diffusion parameter is obtained. A comparison of the accuracy and convergence efficiency of present method with those of PINNs and gPINNs shows that this method can efficiently approximate the large gradient solution of the convective dominant diffusion equation with an accuracy of the order of 10-3 under the condition of unknown boundary layer position. Meanwhile, the present method has more advantages and better performance than PINNs and gPINNs in terms of convergence speed and stability.

Key words: diffusion equation, boundary layer, physical information neural networks, deep learning, convective diffusion

中图分类号:  O241.82

[1] AZIZ K, SETTARI A. Petroleum reservoir simulation[M]. London: Applied Science Publishers, 1979.
[2] ACKERMANN I J, HASS H, MEMMESHEIMER M, et al. Modal aerosol dynamics model for Europe: development and first applications[J]. Atmospheric Environment, 1998, 32(17): 2981-2999. DOI: 10.1016/S1352-2310(98)00006-5.
[3] MORTON K W. Numerical solution of convection-diffusion problems[M]. London: Chapman & Hall, 1996.
[4] SEINFELD J H, PANDIS S N. Atmospheric chemistry and physics: from air pollution to climate change[M]. 2nd ed. Hoboken: Wiley, 2006.
[5] BEAR J. Hydraulics of groundwater[M]. New York: Courier Corporation, 2012.
[6] CLAVERO C, JORGE J C, LISBONA F. A uniformly convergent scheme on a nonuniform mesh for convection-diffusion parabolic problems[J]. Journal of Computational and Applied Mathematics, 2003, 154(2): 415-429. DOI: 10.1016/S0377-0427(02)00861-0.
[7] GE Y B, CAO F J. Multigrid method based on the transformation-free HOC scheme on nonuniform grids for 2D convection diffusion problems[J]. Journal of Computational Physics, 2011, 230(10): 4051-4070. DOI: 10.1016/j.jcp.2011.02.027.
[8] KUMAR V, SRINIVASAN B. An adaptive mesh strategy for singularly perturbed convection diffusion problems[J]. Applied Mathematical Modelling, 2015, 39(7): 2081-2091. DOI: 10.1016/j.apm.2014.10.019.
[9] DU J, CHUNG E. An adaptive staggered discontinuous Galerkin method for the steady state convection-diffusion equation[J]. Journal of Scientific Computing, 2018, 77(3): 1490-1518. DOI: 10.1007/s10915-018-0695-9.
[10] DOUGLAS J J, RUSSELL T F. 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: 10.1137/0719063.
[11] 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.
[12] 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.
[13] 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.
[14] EYMARD R, GALLOUËT T, HERBIN R. Finite volume methods[J]. Handbook of Numerical Analysis, 2000, 7: 713-1018. DOI: 10.1016/S1570-8659(00)07005-8.
[15] 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.
[16] BLECHSCHMIDT J, ERNST O G. Three ways to solve partial differential equations with neural networks: a review[J]. GAMM-Mitteilungen, 2021, 44(2): e202100006. DOI: 10.1002/gamm.202100006.
[17] PANG G F, LU L, KARNIADAKIS G E.fPINNs: fractional physics-informed neural networks[J]. SIAM Journal on Scientific Computing, 2019, 41(4): A2603-A2626. DOI: 10.1137/18M1229845.
[18] JAGTAP A D, KARNIADAKIS G E. Extended physics-informed neural networks(XPINNs): a generalized space-time domain decomposition based deep learning framework for nonlinear partial differential equations[J]. Communications in Computational Physics, 2020, 28(5): 2002-2041. DOI: 10.4208/cicp.OA-2020-0164.
[19] YU J, LU L, MENG X H, 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. DOI: 10.1016/j.cma.2022.114823.
[20] YANG L, MENG X H, KARNIADAKIS G E. B-PINNs: Bayesian physics-informed neural networks for forward and inverse PDE problems with noisy data[J]. Journal of Computational Physics, 2021, 425: 109913. DOI: 10.1016/j.jcp.2020.109913.
[21] JIN X W, CAI S Z, LI H, et al.NSFnets(Navier-Stokes flow nets): Physics-informed neural networks for the incompressible Navier-Stokes equations[J]. Journal of Computational Physics, 2021, 426: 109951. DOI: 10.1016/j.jcp.2020.109951.
[22] JAGTAP A D, KAWAGUCHI K, EM KARNIADAKIS G. Locally adaptive activation functions with slope recovery for deep and physics-informed neural networks[J]. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2020, 476(2239): 20200334. DOI: 10.1098/rspa.2020.0334.
[23] MENG X H, LI Z, ZHANG D K, et al. PPINN:parareal physics-informed neural network for time-dependent PDEs[J]. Computer Methods in Applied Mechanics and Engineering, 2020, 370: 113250. DOI: 10.1016/j.cma.2020.113250.
[24] CHEN Y T, CHANG H B, MENG J, et al. Ensemble neural networks(ENN): a gradient-free stochastic method[J]. Neural Networks, 2019, 110: 170-185. DOI: 10.1016/j.neunet.2018.11.009.
[25] KHARAZMI E, ZHANG Z Q, KARNIADAKIS G E M. hp-VPINNs: variational physics-informed neural networks with domain decomposition[J]. Computer Methods in Applied Mechanics and Engineering, 2021, 374: 113547. DOI: 10.1016/j.cma.2020.113547.
[26] GU Y Q, YANG H Z, ZHOU C.SelectNet: self-paced learning for high-dimensional partial differential equations[J]. Journal of Computational Physics, 2021, 441: 110444. DOI: 10.1016/j.jcp.2021.110444.
[27] DWIVEDI V, SRINIVASAN B. Physics informed extreme learning machine(PIELM)-A rapid method for the numerical solution of partial differential equations[J]. Neurocomputing, 2020, 391: 96-118. DOI: 10.1016/j.neucom.2019.12.099.
[28] SHENG H L, YANG C. PFNN: a penalty-free neural network method for solving a class of second-order boundary-value problems on complex geometries[J]. Journal of Computational Physics, 2021, 428: 110085. DOI: 10.1016/j.jcp.2020.110085.
[29] RAISSI M, YAZDANI A, KARNIADAKIS G E. Hidden fluid mechanics: learning velocity and pressure fields from flow visualizations[J]. Science, 2020, 367(6481): 1026-1030. DOI: 10.1126/science.aaw4741.
[30] MAO Z P, JAGTAP A D, KARNIADAKIS G E. Physics-informed neural networks for high-speed flows[J]. Computer Methods in Applied Mechanics and Engineering, 2020, 360: 112789. DOI: 10.1016/j.cma.2019.112789.
[31] CAI S Z, WANG Z C, WANG S F, et al. Physics-informed neural networks for heat transfer problems[J]. Journal of Heat Transfer, 2021, 143(6): 060801. DOI: 10.1115/1.4050542.
[32] JAGTAP A D, KAWAGUCHI K, KARNIADAKIS G E. Adaptive activation functions accelerate convergence in deep and physics-informed neural networks[J]. Journal of Computational Physics, 2020, 404: 109136. DOI: 10.1016/j.jcp.2019.109136.
[33] VAN DER MEER R, OOSTERLEE C W, BOROVYKH A. Optimally weighted loss functions for solving PDEs with neural networks[J]. Journal of Computational and Applied Mathematics, 2022, 405: 113887. DOI: 10.1016/j.cam.2021.113887.
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