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

Abstract In this article, the bilinear neural network method(BNNM) is used for the first time to investigate the (3+1)-dimensional fourth-order nonlinear equation. Various bilinear neural network models are created by building diverse layers of neural networks. The precise analytical solutions of this equation consists of lump solution, lump and soliton interaction solution, breathing solution, and interference wave solution. To better reveal the dynamic behavior and physical characteristics of these solutions, constructed 3D maps, contour maps, and density maps. The method adopted in this article helps study nonlinear partial differential equations in nonlinear dynamics, oceanography, and fluid mechanics. Finally, we hope these solutions we obtained can explain some nonlinear phenomena of high-order equations.