On implementing Hirota bilinear neural network method and physic-informed of a new fourth-order nonlinear (3+1)-dimensional Ablowitz–Kaup–Newell–Segur equation

Bilinear neural network method and Physic-Informed neural network method have been applied in many aspects of sciences. This paper investigates a class of exact and approximate solutions for a fourth-order nonlinear (3+1)-dimensional Ablowitz–Kaup–Newell–Segur (3AKNS) equation by combining the Hirota bilinear method with a Physics-Informed Neural Network (PINN) framework. First, the governing equation is reduced via a traveling wave transformation, and the Hirota bilinear form is constructed using a logarithmic transformation. Based on this formulation, various analytical solutions, including kink and lump structures, are derived for different parameter cases. To enhance computational efficiency and extend the solution capability, a PINN-based approach is developed to approximate the bilinear function. A softmax-weighted residual loss is introduced to improve convergence and stability during training. Extensive numerical experiments are conducted to analyze the performance of the proposed method in terms of loss convergence, accuracy, and residual decay. The graphical results demonstrate that the proposed framework successfully captures a wide range of nonlinear wave phenomena, including localized lump solutions and smooth kink-type waves. Moreover, the convergence analysis confirms the robustness and competitive accuracy of the method across all considered cases. The integration of analytical and deep learning techniques provides a flexible and efficient tool for studying high-dimensional nonlinear evolution equations. Furthermore, the practical effectiveness of the proposed structure is validated through a fluid flow prediction problem, where the model successfully captures nonlinear spatio-temporal flow patterns and achieves competitive prediction performance.