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.

Accurate and efficient prediction of spatiotemporal variations in the distribution of substances in fluids (SIFs) is crucial for various aspects of fluid mechanics related research and applications, involving for instance, material transport quantification, water quality assessment, and engineering condition analysis. This study proposes a framework for resolving the spatiotemporal distribution of SIFs such as salt and suspended sediment based on water levels and flow velocities. The framework incorporates a deep learning model based on a classic neural operator (DeepONet) architecture, which consists of a feature network and a position network to encode the characteristics of input variables and the problem domain. Numerical simulations were performed to generate the needed datasets. The framework was well-validated by predicting salinity and suspended sediment concentration (SSC) distributions in two idealized cases and a real-word case, demonstrating its efficacy and robustness. Time-series validation further demonstrated the prediction accuracy of the framework. The deep learning model is also capable of enhanced-resolution predictions, enabling the generation of high-resolution spatial distributions of SIFs from low-resolution hydrodynamic data. Both bottom and surface layers of the water column were analyzed, revealing that the mapping relationships between hydrodynamics and SIF distributions can be accurately captured throughout the water column, despite variations in correlation coefficients. Due to these capabilities and advantages, additional data sources can be integrated into the framework in the future, highlighting its considerable potential for broader applications in aquatic environments.