Bilinear feature-enhanced symbolic computation neural network method for solving the (1+1)-dimensional Caudrey–Dodd–Gibbon equation

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.