Lump, breathing inelastic collision phenomena and rogue wave solutions for a extended KP hierarchy-type equation by neural network-based method

This extensive study presents a novel computational framework that effectively investigates the ( 3 + 1 ) -dimensional extended Kadomtsev–Petviashvili (eKP) hierarchy-type equation by combining neural network topologies with a symbolic computing technique. Utilizing various neural network models, including multiple multi-layer topologies, the proposed method enhances both accuracy and computational flexibility. The technique facilitates the generation of diverse solution structures, including localized and wave-like patterns, through the careful manipulation of activation functions and network depth. A deeper understanding of the underlying physical structures of the resulting solutions is provided by a thorough graphical depiction, which includes contour maps, density distributions, and three-dimensional surface graphs, further demonstrating the efficacy of this approach. The novelty and significance of this work lie in the unique integration of symbolic computation methods with neural network architectures to resolve the presented complex ( 3 + 1 ) -dimensional eKP hierarchy-type equation. In contrast to the traditional analytical or numerical methods, this hybrid technique not only ensures greater computational accuracy but also improves adaptability with customized neural structures.