New solutions to the (3+1)-dimensional HB equation using bilinear neural networks method and symbolic ansatz method using neural network architecture

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.