Lump solution, lump and soliton interaction solution, breather solution, and interference wave solution for the (3+1)-dimensional fourth-order nonlinear equation by bilinear neural network method

In this paper, the bilinear neural network method (BNNM) is used for the first time to investigate the (3+1)-dimensional fourth-order nonlinear equation and various bilinear neural network models are created by building diverse layers of neural networks. The precise analytical solutions of this equation include lump solution, lump and soliton interaction solution, breather solution, and interference wave solution. To better reveal the dynamic behavior and physical characteristics of these solutions, we constructed 3D plots, contour plots, and density plots. The method adopted in this paper helps study nonlinear partial differential equations in nonlinear dynamics, oceanography, and fluid mechanics. The obtained precise analytical solutions provide theoretical bases for solving high-order nonlinear equations and advancing research in nonlinear dynamics and their applications. Finally, we hope these solutions will offer new perspectives for interpreting nonlinear phenomena in high-order equations.