Shallow water wave dynamics and novel exact solutions of the (2+1)-dimensional BLMP equation via an improved bilinear neural network method

We investigate the ( 2 + 1 ) -dimensional Boiti–Leon–Manna–Pempinelli (BLMP) equation, a nonlinear dispersive model governing shallow water wave propagation and incompressible fluid dynamics. Applying the improved bilinear neural network method (IBNNM) with composite activation functions, we transform the governing equation into its Hirota bilinear form and construct nine families of exact analytical wave solutions using single-hidden-layer (3-2-1, 3-3-1, 3-4-1) and double-hidden-layer (3-2-2-1, 3-2-3-1) neural-network-type trial functions. The activation functions employed include polynomial, exponential, trigonometric, hyperbolic, and mixed composite forms. The main innovations of this work are: (i) a systematic correspondence is established between neural-network architecture choice and the qualitative class of wave solution obtained from algebraically localized lump-type profiles and breather patterns to periodic wave trains, kink-transition chains, and multi-scale nested oscillations; (ii) double-hidden-layer architectures generate nested functional structures that produce solution families inaccessible to single-layer constructions and to classical methods such as Hirota bilinear, Darboux transformation, and expansion-based approaches; and (iii) the free network parameters are given explicit physical interpretations as wave amplitude, spatial width, propagation velocity, and localization extent. The results provide new analytical benchmarks and physical insight into how the higher-order mixed derivative term Ω x x x y and quadratic nonlinear terms govern wave behavior in shallow water and stratified fluid systems.