Auxiliary Solutions via Dimension Reduction for the (2+1)-Dimensional Calogero–Bogoyavlenskii–Schiff Equation Based on a Bilinear Neural Network Method Coupled with Symbolic Computation

Abstract

Nonlinear partial differential equations formed the mathematical core of nonlinear science. They revealed the intrinsic link between algebraic structures and complex dynamical behaviors. Traditional analytical and numerical methods, however, faced high computational costs, cumbersome derivations, limited accuracy, and poor generality. They especially struggled to solve high-dimensional integrable systems efficiently. To meet these challenges, this study built on traditional neural network methods. We introduced the Hirota bilinear transformation to convert the original nonlinear equation into a bilinear form. We then constructed a bilinear neural network solving procedure deeply coupled with symbolic computation. This strategy effectively suppressed error accumulation caused by random initialization and approximate approximation in traditional numerical iterative methods. Within this framework, this study constructed complete single-hidden-layer and double-hidden-layer neural network models to obtain analytical solutions of the auxiliary equation. By designing network structures with multiple activation functions, we built diverse trial solutions. We also systematically established a coupling adaptation mechanism among activation functions, network depth, and the bilinear transformation. To verify the effectiveness and robustness of the proposed method, we applied it to the Calogero–Bogoyavlenskii–Schiff (CBS) equation. We successfully derived multiple new exact solutions. We further constructed various neural network models and heuristic strategies. This study provided new ideas for neural network solving of integrable systems. It also offered a feasible path for approximation-free exact solutions of high-dimensional nonlinear partial differential equations. The findings held important theoretical value and broad application prospects in fluid dynamics, optical communication, plasma physics, and related engineering fields.