A Robust Modified Gradient-Augmented PINNs Based on Adaptive Weight Loss for Predicting Soliton Solutions of Non-linear Sine-Gordon Equation

This study presents an accurate and robust physics-informed deep learning framework for computing soliton solutions of the \\((2 + 1)\\) -dimensional nonlinear sine-Gordon equation in both damped and undamped regimes. To overcome the limitations of conventional physics-informed neural networks (PINNs), including gradient imbalance, optimization difficulties, and reduced accuracy in the presence of sharp gradients and multi-scale dynamics, we propose an adaptive weight-loss gradient-augmented PINN (AWL-gPINN) formulation. The proposed approach integrates gradient-enhanced physics constraints with an adaptive weight-loss mechanism that dynamically balances the contributions of the governing equation, boundary condition, initial condition, and gradient-enhanced loss terms during training. This formulation improves convergence and prediction accuracy since it is more adaptable and reduces the gradient pathologies that are frequently observed in conventional PINNs. Numerous numerical experiments of sine-Gordon problems involving different soliton structures, such as perturbation of a line soliton, breather solitons on large-scale spatial domains, circular ring solitons, and the superposition of two orthogonal line solitons, reveal that AWL-gPINNs are very reliable in terms of accuracy, stability, and long-time prediction fidelity in comparison to the classical PINNs. Quantitative comparisons based on MSE, RMSE, \\(L_{\\infty }\\) , and \\(L_{2}\\) errors show reductions of one to two orders of magnitude, along with better generalization and smoother convergence. Compared to existing methods in the literature, AWL-PINNs produce superior and accurate outcomes. Additionally, sensitivity analyses under various noise levels and damping parameter adjustments validate the robustness and excellent generalization capabilities of the suggested model. These findings show that the AWL-gPINNs framework offers a scalable and dependable computational method for simulating complex soliton dynamics in nonlinear field theories.