Physics-Informed Deep Learning for Soliton Dynamics and Energy Flow Control in Nonlinear Physical Systems

Abstract

Nonlinear field equations that model high-energy physical systems are frequently hampered by limited integration of deep learning frameworks and the absence of robust hybrid analytical-computational methodologies. In the current research, we overcome such challenges by examining the complex coupled Higgs field equations (CCHFE) by using a novel combination of analytical and deep learning methods. First of all, the Kumar--Malik (KM) approach is used to obtain exact traveling wave solutions such as bright, dark, singular, and periodic soliton structures. These solutions are graphically visualized using high-resolution three-dimensional (3D) surface plots, two-dimensional (2D) cross-sectional profiles, and full heatmap representations, all generated in a Python computational environment. Subsequently, for the first time, an inverse Kumar--Malik physics-informed neural network (IKM-PINN) is proposed for identifying unknown physical parameters and reconstructing soliton dynamics in an inverse-learning setting. The proposed hybrid model incorporates physics-based constraints, analytical soliton profiles, and data-driven learning processes, resulting in highly accurate recovery of bright soliton solutions. A quantitative comparison between the KM solutions and predictions of the IKM-PINN demonstrates excellent agreement resulting in a mean squared error (MSE) of $2 \times 10^{-8}$ and mean absolute error (MAE) of $8.43 \times 10^{-5}$ and root mean squared error (RMSE) of $1.41 \times 10^{-4}$. Furthermore, a comprehensive dynamical and sensitivity analysis is performed for the first time to determine the parametric robustness and stability of the obtained solutions. Finally, the application of the proposed framework is commented on in the context of aerospace energy stabilization, directed energy and defense systems, and energy flow control in high-speed vehicles.