<p>In this paper, the physics-informed neural network (PINN) is applied for the first time to solve the Gerdjikov-Ivanov equation with initial-boundary value conditions. By selecting different types of initial conditions, we successfully derive several localized wave solutions for the Gerdjikov-Ivanov equation, including data-driven soliton solutions, periodic waves, rogue waves, and rogue periodic waves. The results of numerical simulation experiments show that there is only a small margin of error between the data-driven solutions and the exact ones, which verifies that PINN is an effective tool for solving partial differential equations with initial-boundary value conditions. In addition, the propagation phenomena corresponding to these data-driven solutions are analyzed in detail through image simulation. Furthermore, we apply a fast, residue-based attention scheme to improve the accuracy of PINN in learning data-driven localized wave phenomena. Notably, the inverse problems of the Gerdjikov-Ivanov equation are discussed for the first time by applying the PINN method to identify the parameters of the equation based on its soliton, periodic, and rogue wave solutions.</p>