  High resolution simulations of incompressible flows have become routine
across a range of engineering applications. Despite their routine use, due to
the high dimensional parameter space present for most practical applications, a
comprehensive exploration of the available parameter space is often
impractical. In this work, we demonstrate the ability of physics-constrained
deep learning methods to provide an efficient means of exploring
high-dimensional parameter spaces with minimal amounts of data from high
resolution computational fluid dynamic simulations. As a specific application,
we choose the well established problem of a 2D lid driven cavity flow. While
giving an extensive treatment of the classic case of a square cavity, we extend
the analysis to treat an isosceles trapezoid. In so doing, the number of
parameters determining the solution includes not just the Reynolds number, but
also two additional parameters characterizing the geometry of the cavity. Thus,
together with the $\left( x, y \right)$ variation of the flow and pressure in
configuration space, the presence of these three parameters results in the
solution varying in a 5D space. It is shown that in the absence of data,
physics-constrained methods are able to provide an accurate description of the
cavity flow in this 5D space up to intermediate values of the Reynolds number,
but fails to train for sufficiently high Reynolds numbers. In contrast, using a
small quantity of flow data, a single neural network is able to provide an
accurate description for a broad range of Reynolds numbers and cavity
geometries. Once trained, such a model provides a rapid surrogate that can be
used to efficiently explore the 5D space. This 5D surrogate model is
subsequently used to identify critical parameter values for the merger and
splitting of vortices as the Reynolds number and cavity geometry are varied.
