In this article, we focused on the bilinear neural network method to obtain the exact solutions of the (2+1)-dimensional generalized Hirota-Satsuma-Ito-Shallow-Water-Wave-like equation. This method can also be applied to solve other nonlinear partial differential equations (NPDEs). Relying on Hirota bilinear transformation and Bell polynomial theories, we successfully constructed one-hidden-layer and multiple-hidden-layers models with assigned activation functions. Thereby, we derived many exact solutions for this equation including three-wave, rogue wave, and interaction solutions. To intuitively reflect features of these solutions, we depicted their three-dimensional, density, and curve graphs.