Artificial Neural Networks and Simulation of Nonlinear Soliton Solutions of the Modified Benjamin–Bona–Mahony Equation in Nonlinear Optics

In order to investigate the soliton solutions of the third-order nonlinear modified Benjamin–Bona–Mahony equation, this research presents a hybrid analytical and machine-learning methodology. A novel combination of symbolic computation and data-driven modeling is introduced to strengthen the theoretical analysis and improve the simulation capabilities. Next, we propose a Riccati sub-equation neural network (RSENN) framework in which the traveling-wave transformation and Riccati sub-equation structure are incorporated into the neural network model. The proposed RSENN framework accurately approximates the soliton solutions and predicts their spatiotemporal evolution governed by the modified Benjamin–Bona–Mahony equation. The data-driven discovery of the model equation is also performed by estimating the unknown parameters under different noise intensities. The RSENN model is trained using the Levenberg–Marquardt algorithm, and its accuracy and predictive capability are evaluated by comparing its numerical outputs with the exact analytical solutions. The results demonstrate the robustness of the proposed RSENN approach under different levels of noise contamination.