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Numerical Simulation of the Korteweg–de Vries Equation with Machine Learning

Author

Listed:
  • Kristina O. F. Williams

    (Department of Mathematics and Statistics, Air Force Institute of Technology, Dayton, OH 45433, USA)

  • Benjamin F. Akers

    (Department of Mathematics and Statistics, Air Force Institute of Technology, Dayton, OH 45433, USA)

Abstract

A machine learning procedure is proposed to create numerical schemes for solutions of nonlinear wave equations on coarse grids. This method trains stencil weights of a discretization of the equation, with the truncation error of the scheme as the objective function for training. The method uses centered finite differences to initialize the optimization routine and a second-order implicit-explicit time solver as a framework. Symmetry conditions are enforced on the learned operator to ensure a stable method. The procedure is applied to the Korteweg–de Vries equation. It is observed to be more accurate than finite difference or spectral methods on coarse grids when the initial data is near enough to the training set.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:13:p:2791-:d:1175857
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    References listed on IDEAS

    as
    1. Yuexing Bai & Temuer Chaolu & Sudao Bilige, 2021. "Physics Informed by Deep Learning: Numerical Solutions of Modified Korteweg-de Vries Equation," Advances in Mathematical Physics, Hindawi, vol. 2021, pages 1-11, May.
    2. Benjamin Akers & Tony Liu & Jonah Reeger, 2020. "A Radial Basis Function Finite Difference Scheme for the Benjamin–Ono Equation," Mathematics, MDPI, vol. 9(1), pages 1-12, December.
    3. Akers, Benjamin, 2012. "The generation of capillary-gravity solitary waves by a surface pressure forcing," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 82(6), pages 958-967.
    4. Justin Sirignano & Konstantinos Spiliopoulos, 2017. "DGM: A deep learning algorithm for solving partial differential equations," Papers 1708.07469, arXiv.org, revised Sep 2018.
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