  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.
