Solving Shallow Water Equations with Topography using Physics-Informed Neural Networks

Physics-informed neural networks (PINNs) have recently been developed as a novel solution approach for physical problems governed by partial differential equations (PDEs). Compared to purely data-driven methods, PINNs have the advantage of embedding physical constraints in the training process, thus increasing their reliability. Compared to traditional numerical methods for PDEs, PINNs have the advantage of being “meshless”; they are in general less accurate and more computationally expensive, but also more suitable to sparse-data assimilation and to inverse modelling, which is increasing their popularity in many scientific fields. However, hydraulic applications of PINNs in the context of free-surface flows are still in their infancy.

In this work, the effectiveness of PINNs to model one-dimensional free-surface flows over non-horizontal bottom is tested. The governing PDEs are the shallow water equations (SWEs), which represent the mass and momentum conservation in free-surface flows. The inclusion of a spatially variable topography in a meshless method such as PINNs is not trivial. Here, the idea of solving the augmented system of SWEs with topography is exploited. Augmentation consists in treating the bed elevation as a conserved variable (together with water depth and unit discharge) and adding a fictitious equation to the system, which states that this variable is constant in time (i.e., its time derivative is null), while it can be variable in space (its space derivative is included in the bed slope source term). In this way, bed elevation can be easily provided with other initial conditions, and the fixed-bed constraint preserves its value in time.

Different cases of unsteady flows with flat and non-flat bottom are considered, and the accuracy obtained using PINNs with augmented SWEs is checked by comparing PINNs predictions with analytical solutions. Results show that a fair accuracy for depth and velocity can be obtained, even for some challenging test cases such as the dam-break over a bottom step and the planar flow over a parabolic basin (Thacker’s test case). Moreover, it is shown that, if PINNs are applied to a case with horizontal bottom, for which topography is not strictly necessary, similar accuracy and computational time are obtained when PINNs solve standard SWEs or augmented SWEs. It can therefore be concluded that the augmentation of SWEs is a simple but promising strategy to deal with flows over complex bathymetries using PINNs, which paves the way for applications to flows over more realistic topographies.