Soliton solutions and interaction phenomena in oceanography using Hirota bilinear neural network architecture with Bäcklund transformations: A hybrid paradigm

This study demonstrates the ability of hybrid neural networks to produce soliton and interaction solutions for the Itô equation in three-dimensional spaces representing an idealized ocean environment. The hybrid neural network developed utilizes both the Bäcklund transformation approach and the basic structure of a neural network. Several analytical solutions to the Itô equation were obtained by performing hybrid deep learning on specific neural network architectures (i.e., neural network configurations) consisting of 4-2-1, 4-3-1, and 4-4-1 as the trial function in this study. The analytical solutions studied included multi-wave, breather wave, lump cross-kink, rogue wave, two-wave and three-wave, and lump cross-two-wave-kink. In comparison with traditional methods, bilinear Bäcklund transformer neural networks have a number of advantages, including the capability to model solitons more accurately, enabling examples that exhibit significant phase shifts, amplitude retention, and interaction behavior. Bilinear Bäcklund transformer neural networks are also computationally efficient and allow for the modeling of multiple examples of solitons. 3-D and 2-D plots present information regarding the evolutionary behavior and dynamic characteristics of many of the analytical solutions identified in this study. Overall, the analytical solutions provided here demonstrate that hybrid deep learning can be an effective and practical approach to studying nonlinear partial differential equations.