Physics-informed neural networks (PINNs) are increasingly being used in various scientific disciplines. However, dealing with non-stationary physical processes remains a significant challenge in such models, whereas fluid motions are typically non-stationary. In this study, a PINN-based method was designed and optimized to solve non-stationary fluid dynamics with shallow water equations in a polar coordinate system (PINN-SWEP). It was developed and validated with a classic circular basin case that is well-documented in scientific literature. In the validation case, the wind-induced water surface fluctuations are less than 1 cm, posing challenges in modeling. However, our PINN-SWEP model can accurately simulate such tiny water surface fluctuations and resolve complex fluid motions based on limited and sparse data. A boundary discontinuity problem associated with the use of a polar coordinate system is further discussed and improved, thereby enhancing the applicability of PINN in water research. The methodology can provide an alternative solution for numerical or analytical solutions with high accuracy.