Exact Solutions of Higher-Dimensional Integrable Models Using Extended Trial Functions Within a Symbolic Neural Network Computation

This paper investigates two integrable (2+1)-dimensional nonlinear evolution equations, namely the Kairat–II–X-extended (K–II–X-E) model and the Kairat–II-X (K–II–X)-type model, with the objective of deriving exact analytical soliton solutions. These models serve as fundamental prototypes for nonlinear wave dynamics and arise in a broad range of physical contexts, including fluid flows, optical signal propagation, plasma media, and selected biological systems. A (G′/G)-expansion neural networks (ENNs) framework is developed by rigorously coupling the algebraic mechanism of the classical method with the symbolic neural networks. This unified strategy enables the systematic construction of exact analytical solutions directly from the governing equations. Using this framework, exact one- and two-soliton solutions are constructed, along with a broader class of nonlinear wave structures, demonstrating the capability of the method to capture diverse solution dynamics. The resulting solutions are expressed explicitly in trigonometric, hyperbolic, and rational forms. The dynamical features of the obtained solutions are further illustrated through 2D and 3D surface and polar plots, which reveal their localization properties, propagation characteristics, and interaction patterns in higher-dimensional settings. The results show that the (G′/G)-ENNs method provides an efficient and systematic framework for constructing exact solutions of nonlinear evolution equations. This approach offers a promising analytical tool for investigating complex wave phenomena and underscores the potential of neural-network-assisted symbolic techniques in advancing research in nonlinear mathematical physics.