Physics-informed neural network for solution of forward and inverse kinematic wave problems

Abstract

The one-dimensional kinematic wave (KW) model (KWM) is widely used in surface water and water quality hydrology as well as for modeling the movement of traffic on long highways. The KW model involves a first-order nonlinear partial differential equation which is solved numerically for realistic geometric representation, input, and initial and boundary conditions. This paper developed a method to solve the partial differential equation using a neural network structure, called the physics-informed neural network (PINN). The PINN was applied to solve the one-dimensional nonlinear KWM for forward and inverse problems. To enhance the accuracy and convergence and to suppress the spurious oscillation at the steep front, a technique with redistributed collocation points as an improved algorithm of PINN was employed. To deal with the fractional order of the nonlinear advection term, a stop-gradient technique was proposed. For validation, the KWM for steady and unsteady overland flow, and open channel flow without lateral inflow was analyzed for the forward problem. For the inverse problem, the unknown Manning coefficient was correctly computed, demonstrating the capability of the proposed algorithm for the identification of KWM parameters.