IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v12y2024i21p3430-d1512403.html
   My bibliography  Save this article

Solving Viscous Burgers’ Equation: Hybrid Approach Combining Boundary Layer Theory and Physics-Informed Neural Networks

Author

Listed:
  • Rubén Darío Ortiz Ortiz

    (Grupo Ondas, Instituto de Matemáticas Aplicadas, Universidad de Cartagena, Cartagena de Indias 130014, Colombia
    These authors contributed equally to this work.)

  • Oscar Martínez Núñez

    (Grupo Ondas, Instituto de Matemáticas Aplicadas, Universidad de Cartagena, Cartagena de Indias 130014, Colombia
    These authors contributed equally to this work.)

  • Ana Magnolia Marín Ramírez

    (Grupo Ondas, Instituto de Matemáticas Aplicadas, Universidad de Cartagena, Cartagena de Indias 130014, Colombia
    These authors contributed equally to this work.)

Abstract

In this paper, we develop a hybrid approach to solve the viscous Burgers’ equation by combining classical boundary layer theory with modern Physics-Informed Neural Networks (PINNs). The boundary layer theory provides an approximate analytical solution to the equation, particularly in regimes where viscosity dominates. PINNs, on the other hand, offer a data-driven framework that can address complex boundary and initial conditions more flexibly. We demonstrate that PINNs capture the key dynamics of the Burgers’ equation, such as shock wave formation and the smoothing effects of viscosity, and show how the combination of these methods provides a powerful tool for solving nonlinear partial differential equations.

Suggested Citation

  • Rubén Darío Ortiz Ortiz & Oscar Martínez Núñez & Ana Magnolia Marín Ramírez, 2024. "Solving Viscous Burgers’ Equation: Hybrid Approach Combining Boundary Layer Theory and Physics-Informed Neural Networks," Mathematics, MDPI, vol. 12(21), pages 1-30, November.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:21:p:3430-:d:1512403
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/12/21/3430/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/12/21/3430/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Justin Sirignano & Konstantinos Spiliopoulos, 2017. "DGM: A deep learning algorithm for solving partial differential equations," Papers 1708.07469, arXiv.org, revised Sep 2018.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. William Lefebvre & Enzo Miller, 2021. "Linear-quadratic stochastic delayed control and deep learning resolution," Working Papers hal-03145949, HAL.
    2. A. Max Reppen & H. Mete Soner & Valentin Tissot-Daguette, 2022. "Deep Stochastic Optimization in Finance," Papers 2205.04604, arXiv.org.
    3. Sebastian Jaimungal, 2022. "Reinforcement learning and stochastic optimisation," Finance and Stochastics, Springer, vol. 26(1), pages 103-129, January.
    4. Rong Du & Duy-Minh Dang, 2023. "Fourier Neural Network Approximation of Transition Densities in Finance," Papers 2309.03966, arXiv.org, revised Sep 2024.
    5. Gero Junike & Hauke Stier, 2024. "Enhancing Fourier pricing with machine learning," Papers 2412.05070, arXiv.org.
    6. Ali Al-Aradi & Adolfo Correia & Danilo de Frietas Naiff & Gabriel Jardim & Yuri Saporito, 2019. "Extensions of the Deep Galerkin Method," Papers 1912.01455, arXiv.org, revised Apr 2022.
    7. Salah A. Faroughi & Ramin Soltanmohammadi & Pingki Datta & Seyed Kourosh Mahjour & Shirko Faroughi, 2023. "Physics-Informed Neural Networks with Periodic Activation Functions for Solute Transport in Heterogeneous Porous Media," Mathematics, MDPI, vol. 12(1), pages 1-23, December.
    8. Jiequn Han & Ruimeng Hu & Jihao Long, 2020. "Convergence of Deep Fictitious Play for Stochastic Differential Games," Papers 2008.05519, arXiv.org, revised Mar 2021.
    9. Dehghani, Hamidreza & Zilian, Andreas, 2021. "A hybrid MGA-MSGD ANN training approach for approximate solution of linear elliptic PDEs," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 190(C), pages 398-417.
    10. Jesús Fernández-Villaverde & Galo Nuño & Jesse Perla, 2024. "Taming the Curse of Dimensionality: Quantitative Economics with Deep Learning," NBER Working Papers 33117, National Bureau of Economic Research, Inc.
    11. Jialiang Luo & Harry Zheng, 2023. "Deep Neural Network Solution for Finite State Mean Field Game with Error Estimation," Dynamic Games and Applications, Springer, vol. 13(3), pages 859-896, September.
    12. Olivier Bokanowski & Averil Prost & Xavier Warin, 2023. "Neural networks for first order HJB equations and application to front propagation with obstacle terms," Partial Differential Equations and Applications, Springer, vol. 4(5), pages 1-36, October.
    13. Ying Li & Longxiang Xu & Shihui Ying, 2022. "DWNN: Deep Wavelet Neural Network for Solving Partial Differential Equations," Mathematics, MDPI, vol. 10(12), pages 1-35, June.
    14. Li, Wei & Zhang, Ying & Huang, Dongmei & Rajic, Vesna, 2022. "Study on stationary probability density of a stochastic tumor-immune model with simulation by ANN algorithm," Chaos, Solitons & Fractals, Elsevier, vol. 159(C).
    15. Li, Jiaheng & Li, Biao, 2022. "Mix-training physics-informed neural networks for the rogue waves of nonlinear Schrödinger equation," Chaos, Solitons & Fractals, Elsevier, vol. 164(C).
    16. Laura Leal & Mathieu Lauri`ere & Charles-Albert Lehalle, 2020. "Learning a functional control for high-frequency finance," Papers 2006.09611, arXiv.org, revised Feb 2021.
    17. Antoine Jacquier & Zan Zuric, 2023. "Random neural networks for rough volatility," Papers 2305.01035, arXiv.org.
    18. José Alberto Rodrigues, 2024. "Using Physics-Informed Neural Networks (PINNs) for Tumor Cell Growth Modeling," Mathematics, MDPI, vol. 12(8), pages 1-9, April.
    19. Zhang, Jincheng & Zhao, Xiaowei, 2021. "Spatiotemporal wind field prediction based on physics-informed deep learning and LIDAR measurements," Applied Energy, Elsevier, vol. 288(C).
    20. Lukas Gonon, 2022. "Deep neural network expressivity for optimal stopping problems," Papers 2210.10443, arXiv.org.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:12:y:2024:i:21:p:3430-:d:1512403. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.