Physics-informed neural network for diffusive wave model

The diffusive wave model (DWM), a nonlinear second-order simplified form of the shallow water equation, has been widely used in hydraulic, hydrologic and irrigation engineering. Solution of the forward problem of the DWM can be utilized to predict evolution in water levels and discharge. Solution of its inverse problem allows for the identification of crucial parameters (such as Manning coefficient, rainfall intensity, etc.) based on observations. This paper applied the physics-informed neural network (PINN) with novel improvements to solve the DWM for both forward and inverse problems. In the forward problem, compared to traditional numerical methods, PINN was able to predict the evolution at any location. In the inverse problem, PINN provided a simple and efficient solution process. In order to overcome the gradient explosion in the training process caused by the characteristics of the DWM, the stop-gradient technique was adopted to train the neural network. To improve the estimation of DWM parameters, the concept of time division was developed, and a new network structure was proposed. To verify the effectiveness of PINN and its improved algorithm for DWM, seven examples were simulated. The PINN solutions for forward problems were compared with the results obtained by classical numerical methods, while the correct rainfall pattern was identified for the inverse problem.