Physics-Informed Neural Network water surface predictability for 1D steady-state open channel cases with different flow types and complex bed profile shapes

The behavior of many physical systems is described by means of differential equations. These equations are usually derived from balance principles and certain modelling assumptions. For realistic situations, the solution of the associated initial boundary value problems requires the use of some discretization technique, such as finite differences or finite volumes. This research tackles the numerical solution of a 1D differential equation to predict water surface profiles in a river, as well as to estimate the so-called roughness parameter. A very important concern when solving this differential equation is the ability of the numerical model to capture different flow regimes, given that hydraulic jumps are likely to be observed. To approximate the solution, Physics-Informed Neural Networks (PINN) are used. Benchmark cases with different bed profile shapes, which induce different flows types (supercritical, subcritical, and mixed) are tested first. Then a real mountain river morphology, the so-called Step-pool, is studied. PINN models were implemented in Tensor Flow using two neural networks. Different numbers of layers and neurons per hidden layer, as well as different activation functions (AF), were tried. The best performing model for each AF (according to the loss function) was compared with the solution of a standard finite difference discretization of the steady-state 1D model (HEC-RAS model). PINN models show good predictability of water surface profiles for slowly varying flow cases. For a rapid varying flow, the location and length of the hydraulic jump is captured, but it is not identical to the HEC-RAS model. The predictability of the tumbling flow in the Step-pool was good. In addition, the solution of the estimation of the roughness parameter (which is an inverse problem) using PINN shows the potential of this methodology to calibrate this parameter with limited cross-sectional data. PINN has shown potential for its application in open channel studies with complex bed profiles and different flow types, having in mind, however, that emphasis must be given to architecture selection.