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Data-driven rogue waves solutions for the focusing and variable coefficient nonlinear Schrödinger equations via deep
Jiuyun Sun1, Huanhe Dong1, Mingshuo Liu1
1College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China.
Chaos (Woodbury, N.Y.)
|July 19, 2024
Summary
This study uses deep learning, specifically physics-informed memory networks (PIMNs), to accurately solve for rogue wave solutions in nonlinear Schrödinger equations. The method effectively captures complex nonlinear dynamics, advancing AI in solving differential equations.
Area of Science:
- Nonlinear Dynamics
- Computational Physics
- Artificial Intelligence
Background:
- Rogue waves are extreme amplitude events in nonlinear systems.
- Nonlinear Schrödinger (NLS) equations model various wave phenomena, including rogue waves.
- Solving complex differential equations often requires advanced numerical methods.
Purpose of the Study:
- To investigate data-driven rogue wave solutions for focusing and variable coefficient NLS equations.
- To apply physics-informed memory networks (PIMNs) for solving these equations.
- To analyze the impact of network parameters on solution accuracy.
Main Methods:
- Utilizing physics-informed memory networks (PIMNs) for a data-driven approach.
- Solving first- and second-order rogue wave solutions for the focusing NLS equation.
- Solving three deformed rogue wave solutions for the variable coefficient NLS equation.
- Examining the influence of optimization algorithms, network structure, and mesh size.
Main Results:
- PIMNs successfully captured the nonlinear features of rogue wave solutions.
- Demonstrated high accuracy in solving both standard and deformed rogue wave scenarios.
- Numerical experiments confirmed the effectiveness of the deep learning approach.
Conclusions:
- PIMNs are a powerful tool for accurately solving nonlinear Schrödinger equations.
- This deep learning method offers significant potential for understanding rogue wave dynamics.
- The study highlights the advancement of AI in tackling complex partial differential equations.
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