Bogdanov, S;
Shepelsky, D;
Vasylchenkova, A;
Sedov, E;
Freire, PJ;
Turitsyn, SK;
Prilepsky, JE;
(2023)
Phase computation for the finite-genus solutions to the focusing nonlinear Schrödinger equation using convolutional neural networks.
Communications in Nonlinear Science and Numerical Simulation
, 125
, Article 107311. 10.1016/j.cnsns.2023.107311.
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Abstract
We develop a method for retrieving a set of parameters of a quasi-periodic finite-genus (finite-gap) solution to the focusing nonlinear Schrödinger (NLS) equation, given the solution evaluated on a finite spatial interval for a fixed time. These parameters (named “phases”) enter the jump matrices in the Riemann-Hilbert (RH) problem representation of finite-genus solutions. First, we detail the existing theory for retrieving the phases for periodic finite-genus solutions. Then, we introduce our method applicable to the quasi-periodic solutions. The method is based on utilizing convolutional neural networks optimized by means of the Bayesian optimization technique to identify the best set of network hyperparameters. To train the neural network, we use the discrete datasets obtained in an inverse manner: for a fixed main spectrum (the endpoints of arcs constituting the contour for the associated RH problem) and a random set of modal phases, we generate the corresponding discretized profile in space via the solution of the RH problem, and these resulting pairs – the phase set and the corresponding discretized solution in a finite interval of space domain – are then employed in training. The method's functionality is then verified on an independent dataset, demonstrating our method's satisfactory performance and generalization ability.
| Type: | Article |
|---|---|
| Title: | Phase computation for the finite-genus solutions to the focusing nonlinear Schrödinger equation using convolutional neural networks |
| Open access status: | An open access version is available from UCL Discovery |
| DOI: | 10.1016/j.cnsns.2023.107311 |
| Publisher version: | https://doi.org/10.1016/j.cnsns.2023.107311 |
| Language: | English |
| Additional information: | This work is licensed under a Creative Commons Attribution 4.0 International License. The images or other third-party material in this article are included in the Creative Commons license, unless indicated otherwise in the credit line; if the material is not included under the Creative Commons license, users will need to obtain permission from the license holder to reproduce the material. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/ |
| Keywords: | Nonlinear Schrodinger equation, Finite-genus solutions, Nonlinear Fourier transform, Riemann–Hilbert problem, Convolutional neural networks |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Engineering Science UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Engineering Science > Dept of Electronic and Electrical Eng |
| URI: | https://discovery.ucl.ac.uk/id/eprint/10172841 |
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