IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v13y2025i9p1521-d1649452.html
   My bibliography  Save this article

Physics-Informed Neural Networks and Fourier Methods for the Generalized Korteweg–de Vries Equation

Author

Listed:
  • Rubén Darío Ortiz Ortiz

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

  • Ana Magnolia Marín Ramírez

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

  • Miguel Ángel Ortiz Marín

    (Ingeniería de Sistemas y Computación, Universidad Nacional de Colombia, Bogotá 111321, Colombia
    These authors contributed equally to this work.)

Abstract

We conducted a comprehensive comparative study of numerical solvers for the generalized Korteweg–de Vries (gKdV) equation, focusing on classical Fourier-based Crank–Nicolson methods and physics-informed neural networks (PINNs). Our work benchmarks these approaches across nonlinear regimes—including the cubic case ( ν = 3 )—and diverse initial conditions such as solitons, smooth pulses, discontinuities, and noisy profiles. In addition to pure PINN and spectral models, we propose a novel hybrid PINN–spectral method incorporating a regularization term based on Fourier reference solutions, leading to improved accuracy and stability. Numerical experiments show that while spectral methods achieve superior efficiency in structured domains, PINNs provide flexible, mesh-free alternatives for data-driven and irregular setups. The hybrid model achieves lower relative L 2 error and better captures soliton interactions. Our results demonstrate the complementary strengths of spectral and machine learning methods for nonlinear dispersive PDEs.

Suggested Citation

  • Rubén Darío Ortiz Ortiz & Ana Magnolia Marín Ramírez & Miguel Ángel Ortiz Marín, 2025. "Physics-Informed Neural Networks and Fourier Methods for the Generalized Korteweg–de Vries Equation," Mathematics, MDPI, vol. 13(9), pages 1-32, May.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:9:p:1521-:d:1649452
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/13/9/1521/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/13/9/1521/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. 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.
    2. Kristina O. F. Williams & Benjamin F. Akers, 2023. "Numerical Simulation of the Korteweg–de Vries Equation with Machine Learning," Mathematics, MDPI, vol. 11(13), pages 1-14, June.
    3. Kong, Desong & Xu, Yufeng & Zheng, Zhoushun, 2019. "A hybrid numerical method for the KdV equation by finite difference and sinc collocation method," Applied Mathematics and Computation, Elsevier, vol. 355(C), pages 61-72.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Leonid Serkin & Tatyana L. Belyaeva, 2025. "Physics-Informed Neural Networks for Higher-Order Nonlinear Schrödinger Equations: Soliton Dynamics in External Potentials," Mathematics, MDPI, vol. 13(11), pages 1-28, June.

    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. Leonid Serkin & Tatyana L. Belyaeva, 2025. "Physics-Informed Neural Networks for Higher-Order Nonlinear Schrödinger Equations: Soliton Dynamics in External Potentials," Mathematics, MDPI, vol. 13(11), pages 1-28, June.
    2. Kohnesara, Sima Molaei & Firoozjaee, Ali Rahmani, 2023. "Numerical solution of Korteweg–de Vries equation using discrete least squares meshless method," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 206(C), pages 65-76.

    More about this item

    Keywords

    ;
    ;
    ;
    ;
    ;
    ;

    Statistics

    Access and download statistics

    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:13:y:2025:i:9:p:1521-:d:1649452. 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.