Exploring soliton, breather and lump structures in the (4+1)-dimensional KdV-CBS equation with the bilinear neural network method

This work focuses on the [Formula: see text]-dimensional Korteweg-de Vries–Calogero–Bogoyavlenskii–Schiff equation, analyzed through the recently introduced bilinear neural network method. By implementing multiple neural network architectures with both single- and double-hidden layers (5-2-1, 5-3-1, 5-4-1, 5-2-4-1, and 5-3-4-1) in combination with diverse activation functions, a wide class of analytical solutions is derived. The obtained results are illustrated using three-dimensional surface plots alongside their two-dimensional density profiles, highlighting a spectrum of nonlinear wave phenomena such as bell-shaped waves, kink-antikink interactions, rogue waves, breather-type patterns, and lump structures. The findings confirm that the bilinear neural network method offers an effective and robust computational strategy for addressing higher-dimensional nonlinear evolution equations.