Volume 44 Issue 1
Jan.  2023
Turn off MathJax
Article Contents
LIN Yunyun, ZHENG Supei, FENG Jianhu, JIN Fang. Diffusive Regularization Inverse PINN Solutions to Discontinuous Problems[J]. Applied Mathematics and Mechanics, 2023, 44(1): 112-122. doi: 10.21656/1000-0887.430010
Citation: LIN Yunyun, ZHENG Supei, FENG Jianhu, JIN Fang. Diffusive Regularization Inverse PINN Solutions to Discontinuous Problems[J]. Applied Mathematics and Mechanics, 2023, 44(1): 112-122. doi: 10.21656/1000-0887.430010

Diffusive Regularization Inverse PINN Solutions to Discontinuous Problems

doi: 10.21656/1000-0887.430010
  • Received Date: 2022-01-14
  • Accepted Date: 2022-04-27
  • Rev Recd Date: 2022-03-17
  • Available Online: 2022-12-02
  • Publish Date: 2023-01-15
  • It is of great importance to numerically capture discontinuities for the numerical solutions to hyperbolic conservation laws equations. The PINN (physics-informed neural networks) was used to solve the forward problem of the hyperbolic conservation laws equations, with the diffusion term added, which is difficult to determine and needs to be obtained through high-cost trial calculation. To capture the discontinuous solutions and save calculation costs, the equation was regularized through addition of diffusive terms. Then the regularized equation was incorporated into the loss function, and the exact solutions or reference solutions to the conservation laws equations were used as the training set to learn the diffusion coefficients, and the solutions at different moments were predicted. Compared with that of the PINN method for solving forward problems, the resolution of discontinuous solutions was improved, and the trouble of massive trial calculation was avoided. Finally, the feasibility of the algorithm was verified by 1D and 2D numerical experiments. The numerical results show that, the new algorithm has better ability to capture discontinuities, produces no spurious oscillations and no screed phenomena. Additionally, the diffusive coefficients obtained with the new algorithm make a reference to construct the classic numerical scheme.

  • loading
  • [1]
    JIN X, CAI S, 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
    BASDEVANT C, DEVILLE M, HALDENWANG P, et al. Spectral and finite difference solutions of the Burgers equation[J]. Computers & Fluids, 1986, 14(1): 23-41. doi: 10.1016/0045-7930(86)90036-8
    郑素佩, 王令, 王苗苗. 求解二维浅水波方程的移动网格旋转通量法[J]. 应用数学和力学, 2020, 41(1): 42-53

    ZHENG Supei, WANG Ling, WANG Miaomiao. Solution of 2D shallow water wave equation with the moving grid rotating-invariance method[J]. Applied Mathematics and Mechanics, 2020, 41(1): 42-53.(in Chinese)
    贾豆, 郑素佩. 求解二维 Euler 方程的旋转通量混合格式[J]. 应用数学和力学, 2021, 42(2): 170-179

    JIA Dou, ZHENG Supei. A hybrid scheme of rotational flux for solving 2D Euler equations[J]. Applied Mathematics and Mechanics, 2021, 42(2): 170-179.(in Chinese)
    PSICHOGIOS D C, UNGAR L H. A hybrid neural network-first principles approach to process modeling[J]. AICHE Journal, 1992, 38(10): 1499-1511. doi: 10.1002/aic.690381003
    LAGARIS I E, LIKAS A, FOTIADIS D I. Artificial neural networks for solving ordinary and partial differential equations[J]. IEEE Transactions on Neural Networks, 1998, 9(5): 987-1000. doi: 10.1109/72.712178
    高普阳, 赵子桐, 杨扬. 基于卷积神经网络模型数值求解双曲型偏微分方程的研究[J]. 应用数学和力学, 2021, 42(9): 932-947

    GAO Puyang, ZHAO Zitong, YANG Yang. Numerical solution of hyperbolic partial differential equations based on convolutional neural network mode[J]. Applied Mathematics and Mechanics, 2021, 42(9): 932-947.(in Chinese)
    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(C): 686-707.
    LU L, MENG X, MAO Z, et al. DeepXDE: a deep learning library for solving differential equations[J]. Society for Industrial and Applied Mathematics, 2021, 63(1): 208-228.
    MAO Z, 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
    LEFLOCH P G, MISHRA S. Numerical methods with controlled dissipation for small-scale dependent shocks[J]. Acta Numerica, 2014, 23: 743-816. doi: 10.1017/S0962492914000099
    EL G A, HOEFER M A, SHEARER M. Dispersive and diffusive-dispersive shock waves for nonconvex conservation laws[J]. SIAM Review, 2017, 59(1): 3-61. doi: 10.1137/15M1015650
    CLAMOND D, DUTYKH D. Non-dispersive conservative regularisation of nonlinear shallow water (and isentropic Euler equations)[J]. Communications in Nonlinear Science and Numerical Simulation, 2018, 55: 237-247. doi: 10.1016/j.cnsns.2017.07.011
    MINBASHIAN H, GIESSELMANN J. Deep learning for hyperbolic conservation laws with non-convex flux[J]. Proceedings in Applied Mathematics and Mechanics, 2021, 20: e202000347.
    LAX P. Shock waves and entropy[C]//Proceedings of a Symposium Conducted by the Mathematics Research Center, the University of Wisconsin-Madison. New York: Academic Press, 1971: 603-634.
    BAYDIN A G, PEARLMUTTER B A, RADUL A A, et al. Automatic differentiation in machine learning: a survey[J]. Journal of Machine Learning Research, 2018, 18: 1-43.
    KINGMA D P, BA J. Adam: a method for stochastic optimization[Z/OL]. (2017-01-30)[2022-03-17].https://arxiv.org/abs/1412.6980.
    LIU D C, NOCEDAL J. On the limited memory BFGS method for large scale optimization[J]. Mathematical Programming, 1989, 45(1): 503-528.
    TADMOR E. Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems[J]. Acta Numerica, 2003, 12: 451-512. doi: 10.1017/S0962492902000156
  • 加载中


    通讯作者: 陈斌, bchen63@163.com
    • 1. 

      沈阳化工大学材料科学与工程学院 沈阳 110142

    1. 本站搜索
    2. 百度学术搜索
    3. 万方数据库搜索
    4. CNKI搜索


    Article Metrics

    Article views (86) PDF downloads(35) Cited by()
    Proportional views


    DownLoad:  Full-Size Img  PowerPoint