On Deploying Bilinear Neural Network Method to Various Solutions of (3+1)-dimensional Potential Yu-Toda-Sasa-Fukuyama Equation

This paper investigates the (3+1)-dimensional potential Yu-Toda-Sasa-Fukuyama (YTSF) equation using the Bilinear Neural Network Method (BNNM). This novel hybrid framework integrates Hirotas bilinear formalism with neural network modeling. By reformulating the YTSF equation into its bilinear form and embedding this structure into the BNNM architecture, multiple classes of exact analytical solutions are derived, including kink-type, periodic, and rational forms. The proposed method yields closed-form solutions that preserve mathematical rigor while improving computational efficiency for high-dimensional nonlinear evolution equations. The results demonstrate the effectiveness of the BNNM in generating diverse solution structures for the YTSF equation, providing potential applications in fluid dynamics, plasma physics, and other nonlinear physical models.