Data-driven soliton solutions for the perturbed modified Gardner equation in stratified oceanic fluids via orthogonal neural network

In this study, we explore data-driven soliton dynamics and identify unknown parameters in the perturbed modified Gardner equation using an orthogonal neural network (ONN). First, the new modified generalized exponential rational function method generates a comprehensive dataset of exact solutions. The novelty lies in combining analytical techniques with an ONN framework, benefiting from both mathematical interpretability and machine learning predictive capability. Using a traveling-wave transformation, the governing PDE is reduced to a third-order ODE. The unknown wave profile is approximated using an ONN with Gegenbauer polynomials as activation functions. Operational matrices of derivatives transform differential operators into algebraic forms, yielding a system of nonlinear algebraic equations for the network weights, solved iteratively. To validate accuracy, several exact soliton solutions are compared with ONN predictions. The model’s performance is evaluated using root mean square error, maximum absolute error, pointwise absolute error, mean absolute error, and convergence with respect to neuron count. The extremely low errors and close agreement between exact and predicted solitons confirm the effectiveness and reliability of the proposed ONN framework. This approach provides improved prediction with computational efficiency, highlighting its potential as a robust framework for solving complex nonlinear wave equations in stratified oceanic fluids and mathematical physics.