Fast and reliable prediction of riverine flow velocities is important in many applications, including flood risk management. The shallow water equations (SWEs) are commonly used for prediction of the flow velocities. However, accurate and fast prediction with standard SWE solvers is challenging in many cases. Traditional approaches are computationally expensive and require high-resolution riverbed profile measurement ( bathymetry) for accurate predictions. As a result, they are a poor fit in situations where they need to be evaluated repetitively due, for example, to varying boundary condition (BC), or when the bathymetry is not known with certainty. In this work, we propose a two-stage process that tackles these issues. First, using the principal component geostatistical approach (PCGA) we estimate the probability density function of the bathymetry from flow velocity measurements, and then we use multiple machine learning algorithms to obtain a fast solver of the SWEs, given augmented realizations from the posterior bathymetry distribution and the prescribed range of BCs. The first step allows us to predict flow velocities without direct measurement of the bathymetry. Furthermore, the augmentation of the distribution in the second stage allows incorporation of the additional bathymetry information into the flow velocity prediction for improved accuracy and generalization, even if the bathymetry changes over time. Here, we use three solvers, referred to as PCA-DNN (principal component analysis-deep neural network), SE (supervised encoder), and SVE (supervised variational encoder), and validate them on a reach of the Savannah river near Augusta, GA. Our results show that the fast solvers are capable of predicting flow velocities with good accuracy, at a computational cost that is significantly lower than the cost of solving the full boundary value problem with traditional methods.
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