Simulation of the Mzymta River Flow Using the Regularized Physics-Informed Neural Networks

This study considers the simulation of hydrological processes in the Mzymta River (Russian Federation) using regularized Physics-Informed Neural Networks (PINNs). The Mzymta is the longest river in the Russian Federation and flows into the Black Sea. To formulate the boundary value problem, data on river topography (bottom profile) and hydrological measurements (time series of water level and discharge) from the gauging stations near the settlements of Rosa Khutor, Gorki-Gorod, Krasnaya Polyana, and Kazachiy Brod were collected. This study considers two modeling scales: the full-scale problem and a local-scale formulation for a short river segment, characterized by bed slope flattening, the development of a backwater–drawdown profile, and the occurrence of a hydraulic jump. The Saint-Venant system of equations was considered as a mathematical model. To solve this problem, we implemented PINNs. Two problem statements were considered: stationary and non-stationary. The full-scale simulation revealed that at the entire river scale, the flow rapidly establishes a normal depth profile, with all local perturbations being smoothed out by friction and thus indistinguishable at this resolution. In the unsteady case, the flow profile undergoes perturbations that propagate downstream in response to the time-varying sinusoidal inlet discharge. These perturbations propagate downstream at a finite speed, which is determined by the characteristic velocities of the shallow-water system. The local-scale formulation allowed us to capture the deviation of the flow profile from the normal depth and to study the backwater–drawdown problem, which involves the formation of a hydraulic jump both in stationary and non-stationary formulations. The formation of a hydraulic jump required the introduction of regularization into the loss function, since in the vicinity of the jump, the solution ceases to be smooth and experiences a discontinuity. To capture the discontinuity, we applied a method of relaxing the momentum equation term near the jump region. To enhance the sharpness of the jump, we additionally introduced a term that minimizes the width of the jump region. To properly account for all terms in the loss function, adaptive weighting of the coefficients was employed. Verification of the PINN against the fourth-order Runge–Kutta method is performed for the steady-state case, for both smooth solutions and solutions involving a hydraulic jump. The unsteady full-scale formulation is validated against the measured hydrograph between the Krasnaya Polyana and Kazachiy Brod gauging stations. A sensitivity analysis with respect to the river width and the Manning coefficient shows that these parameters strongly affect the depth profile. As a direction for future work, we outline an inverse formulation for the joint identification of the river width and Manning coefficient distributions along the river.