This study examines the Boussinesq equation, which is a nonlinear partial differential equation used to describe long wave propagation in shallow water and has broader applications, including nonlinear lattice waves, vibrations in nonlinear strings, and ion sound waves in plasma. The Boussinesq equation provides an insight into the nonlinear long wave propagation behavior in shallow water by taking wave phase into account. Its versatility extends its utility beyond fluid dynamics to various physical phenomena. By providing specific activation functions in the \(``2-3-1''\) and \(``2-5-1''\) neural network models, respectively, the generalized lump solution and the precise analytical solutions are produced using the bilinear neural network approach. These analytical solutions, together with the related rogue waves, dark soliton, and bright soliton, are derived using symbolic computation. These findings fill in the gaps in the current research about the Boussinesq equation. The dynamical properties of these waves are displayed on three-dimensional, contour, density, and two-dimensional graphs. The response of the wave solution to different values of wave speed in relation to the wave phase it contains has been described with the help of wave intensity. In addition, the advantages and disadvantages of the layers used in the analytical technique to generate solutions have been discussed. The efficient techniques employed in this research are useful for studying the nonlinear differential equations in one-dimensional nonlinear lattice waves, vibrations in a nonlinear string, and ion sound waves in plasma.
