Rogue wave, lump, kink, periodic and breather-like solutions of the (2+1)-dimensional KdV equation

In this paper, the ([Formula: see text])-dimensional KdV equation is investigated by using the bilinear neural network method (BNNM). We construct six neural network models, extending beyond single hidden layer models to create deeper and broader network structures (e.g. [3-3-1], [3-4-1], [3-1-3-1], [3-4-1-1], [3-2-2-1] and [3-2-3-1-1] models). Introducing specific activation functions into the neural network model enables the generation of various test functions, resulting in novel solutions for equations that include rogue wave solutions, lump-kink solutions, periodic soliton solution, breather-like solutions and lump solutions. The physical properties of these novel solutions are vividly depicted through three-dimensional plots, density plots, and curve plots. The findings contribute to a better understanding of the propagation behavior of shallow water waves.