Physics-Informed Neural Networks: A Novel Framework for Solving 1D Saint-Venant Equations

The Saint-Venant equations are extensively employed to model water flow in channels, particularly when a comprehensive analysis is necessary. This study presents a mesh-free approach utilizing Physics-Informed Neural Networks (PINNs) to address the 1D Saint-Venant equations under diverse initial and boundary conditions. PINNs provide substantial benefits compared to conventional hydrodynamic models by enabling predictions at any location within the computational domain without the necessity for predefined computational points or cross-sections. This versatility is especially advantageous for applications necessitating elevated spatial resolution or dynamic adaptability. In contrast to traditional machine learning (ML) methods, Physics-Informed Neural Networks (PINNs) do not necessitate labelled data. Their loss function integrates the residual error of the governing partial differential equations (PDEs) with initial and boundary conditions, thereby ensuring predictions that are physically consistent. This addresses the interpretability deficit frequently linked to machine learning models. The constructed PINNs architecture was evaluated on four test cases that exemplify various channel geometries and flow conditions. The first scenario pertains to a horizontal bed exhibiting a constant upstream velocity. The second case analyses a rectangular channel with a constant slope and dynamic inflow, whereas the third and fourth cases comprise channels with changing slopes and widths. These scenarios reflect real-world water transport channels. In cases 1 and 2, the maximum depth error was ±0.08 m relative to numerical solutions, with the most significant errors occurring at the points of initial water arrival. In cases 3 and 4, the maximum depth errors were ±0.2 m and ±0.4 m, respectively. These findings indicate that PINNs can accurately reproduce numerical solutions without the necessity of a computational mesh. The adaptability of PINNs in sampling collocation points eliminates the necessity for re-simulating the model when results are needed at new locations. This study underscores the efficacy of PINNs for real-time water resource management and flood forecasting, particularly where conventional methods may be computationally expensive or inflexible. Future research will investigate the expansion of the PINNs framework to encompass higher-dimensional Saint-Venant equations and the incorporation of stochastic inputs to address uncertainties in flow conditions.