River Surface Velocity and Discharge Estimation Using Optical Flow and Unlabeled Physics-Informed Neural Networks

Quantifying river surface velocity and discharge is essential for flood control and mitigation. Traditional contact measurement methods are capable of providing precise results, yet they demand considerable manpower and material resources and face implementation challenges in flood seasons. Image velocimetry methods have attracted extensive attention due to their low cost, simplicity in operation, and safety. However, most of them lack a physical basis and interpretability. This paper introduces a river flow estimation algorithm combined with Physics-Informed Neural Networks (PINNs). The introduction of the convection–diffusion equation based on optical flow enables the model to better fit the flow characteristics of water and provides stronger physical support for the measurement results. The adoption of this equation as the loss function and the introduction of multiple scenarios eliminate the need for labeled data in the PINNs training process. The experimental results in both artificial and natural river channels demonstrate that the relative errors of the discharge measured by the proposed method are 0.66% and −1.75%, and the relative errors of the mean velocity are 0.64% and −2.33%. Compared with other methods, the proposed method exhibits superior performance.