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Coarse-Gridded Simulation of the Nonlinear Schrödinger Equation with Machine Learning

Author

Listed:
  • Benjamin F. Akers

    (Department of Mathematics, Air Force Institute of Technology, 2950 Hobson Way, Wright-Patterson AFB, Dayton, OH 45433, USA)

  • Kristina O. F. Williams

    (Department of Mathematics, Air Force Institute of Technology, 2950 Hobson Way, Wright-Patterson AFB, Dayton, OH 45433, USA)

Abstract

A numerical method for evolving the nonlinear Schrödinger equation on a coarse spatial grid is developed. This trains a neural network to generate the optimal stencil weights to discretize the second derivative of solutions to the nonlinear Schrödinger equation. The neural network is embedded in a symmetric matrix to control the scheme’s eigenvalues, ensuring stability. The machine-learned method can outperform both its parent finite difference method and a Fourier spectral method. The trained scheme has the same asymptotic operation cost as its parent finite difference method after training. Unlike traditional methods, the performance depends on how close the initial data are to the training set.

Suggested Citation

  • Benjamin F. Akers & Kristina O. F. Williams, 2024. "Coarse-Gridded Simulation of the Nonlinear Schrödinger Equation with Machine Learning," Mathematics, MDPI, vol. 12(17), pages 1-10, September.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:17:p:2784-:d:1474051
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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. 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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