The fifth-order dispersion nonlinear wave (1+1)-dimensional Caudrey–Dodd–Gibbon equation is a classic model describing soliton phenomena in fields such as plasma magnetosonic waves and optical fiber light pulses, and its exact solution is of great importance for revealing the laws of nonlinear wave motion. In this paper, an integrated framework combining bilinear polynomial feature enhancement, symbolic computation constraints, and neural network learning is proposed. Bilinear polynomial features such as $x^2$, $t^2$, and $xt$ are introduced to break through the input limitation of original variables, broaden the boundary of the model in capturing nonlinear interactions between variables, and reduce errors caused by insufficient feature information. Symbolic computation is applied to the bilinear transformation derivation and conservation law analysis of the (1+1)-dimensional Caudrey–Dodd–Gibbon Equation to provide mathematical structure constraints for the neural network, and a collaborative mechanism of "symbolic reasoning guiding numerical learning" is constructed to improve the interpretability of the model. This framework breaks down the barriers between traditional numerical and pure neural network methods, realizes efficient and accurate solution of the (1+1)-dimensional Caudrey–Dodd–Gibbon equation, and provides a new path for the study of exact solutions of high-dimensional, variable-coefficient, and strongly nonlinear partial differential equations.

This study introduces a novel hybrid computational framework that synergistically integrates neural networks with symbolic computation to address nonlinear partial differential equations (PDEs). By combining the robust nonlinear approximation capabilities of neural networks with the analytical precision of symbolic computation, our proposed symbolic ansatz method using neural network architecture (SANNA) achieves superior accuracy and efficiency compared to conventional numerical techniques. With in this framework, we design three distinct neural network architectures—each incorporating varied trial functions—and further integrate the bilinear neural network method (BNNM). To validate the effectiveness of our methodology, we apply it to the (3+1)-dimensional HB equation, a prototypical nonlinear model with significance in soliton theory and wave dynamics. The approach yields multiple novel analytical solutions, including periodic traveling waves and strongly localized nonlinear modes, all exhibiting clear mathematical interpretability and physical relevance. These results highlight the method's potential for applications in fluid dynamics, ocean engineering, and geophysical flow modeling.

This study proposed a novel symbolic computing algorithm based on neural networks for solving the (3+1)–dimensional Jimbo-Miwa equation. By constructing a direct neural network model and integrating neural networks with symbolic computing, activation functions were assigned to the neurons in the hidden layer of the neural network. After deriving the trial function, symbolic computing using Maple was employed to obtain the exact analytical solution of the equation. Our innovative method effectively avoids the reliance on large datasets and low computational efficiency of traditional methods. Based on this improved method, we have constructed single-hidden-layer and double-hidden-layer neural network models to solve the equation's exact solutions, and successfully obtained breather solutions, shock wave solutions, and lump solutions. The successful solution of the equation in this study fully demonstrates the efficiency of the constructed framework and indicates its promising application prospects in other important nonlinear partial differential equation fields.