Physics-Informed Neural Networks for Steady-State Weir Flows Using the Serre–Green–Naghdi Equations

Abstract

This paper presents physics-informed neural networks (PINNs) to approximate the Serre–Green–Naghdi equations (SGNEs) that model steady-state weir flows. Four PINNs are proposed to solve the forward problem and three types of inverse problem. For the forward problem in which continuous and smooth beds are available, we constructed PINN 1 to predict the water depth profile over a weir. Good agreements between the PINN 1 solutions and experimental data demonstrated the capability of PINN 1 to resolve the steady-state weir flows. For the inverse problems with input discretized beds, PINN 2 was designed to output both the water depth profile and the bed profile. The free-surface profiles based on the PINN 2 solutions were in good agreement with the experimental data, and the reconstructed bed profiles of PINN 2 agreed well with the input discretized beds, demonstrating that PINN 2 can reproduce weir flows accurately when only discretized beds are available. For the inverse problems with input measured free surface, PINN 3 and PINN 4 were built to output both the free-surface profile and the bed profile. The output free-surface profiles of PINN 3 and PINN 4 showed good agreement with the experimental data. The inferred bed profiles of PINN 3 agreed generally well with the analytical weir profile or the control points of the weir profile, and the inferred bed profiles of PINN 4 were in good agreement with the analytical weir profile for the investigated test case. These indicate that the proposed PINN 3 and PINN 4 can satisfactorily infer weir profiles. Overall, PINNs are comparable to the traditional numerical models for forward problems, but they can resolve the inverse problems which cannot be solved directly using traditional numerical models. There has been tremendous progress in solving governing equations using traditional numerical methods. However, when solving inverse problems, traditional numerical methods usually are time consuming and require new algorithms. Most importantly, traditional numerical methods are unable to resolve problems with missing or noisy initial and boundary conditions. Compared with traditional numerical methods, physics-informed neural networks implement a mesh-free algorithm and are effective and efficient for inverse and even ill-posed problems. Physics-informed neural networks integrate physical governing equations and relevant data, e.g., initial and boundary conditions or measured data, to infer unknown variables for forward and inverse problems, and have been applied to solve various types of governing equations. To the best of our knowledge, no PINNs have been presented to solve weir flows. This paper proposes physics-informed neural networks to solve forward and inverse weir flows. Research findings indicate that physics-informed neural networks are comparable to traditional numerical methods for the forward problem and are capable of resolving the inverse problems in which the discretized bed elevations or the measured free surface are available.