Machine learning-assisted modeling of nonlinear dispersive waves in ocean engineering

Artificial neural networks have been found to be useful computational models capable of handling nonlinear problems. Such problems are quite common in ocean engineering. Wave motion in shallow and dispersive media is essentially a nonlinear process with strong dispersive actions, as experienced in near-coastal areas, estuaries, and the continental shelf. Nonlinear wave interactions are of paramount importance in the study of long wave propagation, tidal bores, internal waves, and wave transformation phenomena. This paper examines a detailed study of the (1+1)-dimensional modified Benjamin-Bona-Mahony equation, which is one of the basic nonlinear evolution equations and has been widely applied to describe the propagation of long waves in the dispersive oceanic environment. The hybrid method of using the Riccati sub-equation neural networks is introduced to derive both the exact and approximate solutions. Specifically, the method starts by considering the (1+1)-dimensional modified Benjamin-Bona-Mahony equation. The architecture of 2-2-2-1 Riccati sub-equation neural network is designed, which has two input neurons for spatial and temporal variables, three hidden layers with two neurons in each layer and activation function designed based on Riccati sub-equation, and one output neuron for approximating wave profile. The proposed framework has the ability to capture nonlinear wave phenomena such as solitary waves, periodic wave trains, and their interactions due to its inherent property of incorporating Riccati sub-equation into the neural network architecture. The accuracy and efficacy of the proposed framework are validated through various numerical simulations and graphical analyses.