In this study, we proposed an improved physics-informed neural network (PINNs) for numerical simulation of localized wave solutions of partial differential equations (PDEs). Solving localized wave solutions of PDEs is an essential research topic in nonlinear science with significant research value, and related theory has been employed to many fields. Numerous researchers have conducted a series of studies for localized wave solutions of PDEs. So far, PINNs have been successfully applied to solve various types of complex PDEs and physical simulations, showing great potential for application and receiving much attention from researchers. The innovation of this study is to introduce an adaptive learning method to dynamically adjust the loss function weights of the gradient-enhanced PINNs for the localized wave simulation problem. The improved PINNs not only embed the constraints of PDEs, but also add the constraints of gradient information, thus further enriching the physical constraints of the neural network model. In addition, the weight coefficients of the loss function are updated by an adaptive learning method to dynamically adjust the proportion of each constraint term in the loss function to speed up the training speed. We applied the improved PINNs to perform a large number of experiments that effectively reproduce the dynamic behavior of localized waves. We chose the Boussinesq equation and the nonlinear Schrödinger (NLS) equation for the study, respectively, and evaluated the accuracy of the localized wave simulation results by error analysis. The experimental results show that the improved PINNs significantly outperform the traditional PINNs with shorter training time and more accurate prediction results.