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

Benjamin F. Akers () and Kristina O. F. Williams
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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

Mathematics, 2024, vol. 12, issue 17, 1-10

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.

Keywords: machine learning; nonlinear Schrödinger equation; coarse grid (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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