A novel hybrid analytical–PINNs approach for data-driven soliton dynamics and parameter discovery in the Kairat-II-X equation

This study presents a hybrid analytical and machine learning framework to investigate soliton dynamics in the recently proposed Kairat-II-X (K-II-X) equation, a model unifying features of the Kairat-II and Kairat-X systems. The equation captures geometric properties of differential curves relevant to physical and engineering phenomena. In the first phase, we employ novel polynomial expansion neural networks (PENNs), embedding trial functions into neural architectures where neurons represent polynomial–Riccati operator. This enables the generation of novel analytical soliton structures, including hyperbolic, trigonometric, exponential, and rational functions, in both single-soliton and two-soliton forms. In the second phase, we apply physics-informed neural networks (PINNs) with parameter regularization for numerical soliton solutions and parameter identification. The PINNs framework accurately captures one-kink, two-kink, and periodic dynamics, demonstrating strong agreement with analytical benchmarks and low error metrics. Furthermore, the method effectively recovers nonlinear coefficients from noisy data, highlighting robustness in parameter inference. This integrated symbolic–numeric approach bridges analytical solution construction and machine learning modeling, enhancing soliton solution discovery and providing a versatile platform for simulating complex nonlinear wave phenomena. The framework holds promising implications for applied mathematics, physics, and engineering.