<p>This study explores the rich dynamical behaviors exhibited by the Gerdjikov-Ivanov equation, a classical integrable system with broad physical applications. Firstly, through the analysis of modulation instability theory, we rigorously derived the conditions for the existence of rogue wave solutions, laying a theoretical foundation for subsequent numerical research. At the methodological level, the innovative use of physical information neural network methods has the bottleneck of relying on a large amount of data compared to traditional neural networks. This framework only requires a small number of training samples to achieve high-precision solutions and has strong generalization ability to adapt to different solution types by modifying equation constraints. Based on this method, we predicted and analyzed the dynamic behavior of the generalized Gerdjikov-Ivanov equation from one soliton to three solitons. In addition, by designing controlled variable experiments, the influence of neuron arrangement order on the evolution process of first-order rogue waves and second-order rogue waves in the Gerdjikov-Ivanov equation was analyzed in depth. Numerical experiments have confirmed that this method can not only accurately reproduce the propagation characteristics of classical soliton solutions, but also effectively capture the transient burst characteristics of rogue waves under specific neuron orders, providing new ideas for studying such complex nonlinear phenomena.</p>