The use of reduced-order models (ROMs) in physics-based modeling and
simulation almost always involves the use of linear reduced basis (RB) methods
such as the proper orthogonal decomposition (POD). For some nonlinear problems,
linear RB methods perform poorly, failing to provide an efficient subspace for
the solution space. The use of nonlinear manifolds for ROMs has gained traction
in recent years, showing increased performance for certain nonlinear problems
over linear methods. Deep learning has been popular to this end through the use
of autoencoders for providing a nonlinear trial manifold for the solution
space. In this work, we present a non-intrusive ROM framework for steady-state
parameterized partial differential equations (PDEs) that uses convolutional
autoencoders (CAEs) to provide a nonlinear solution manifold and is augmented
by Gaussian process regression (GPR) to approximate the expansion coefficients
of the reduced model. When applied to a numerical example involving the steady
incompressible Navier-Stokes equations solving a lid-driven cavity problem, it
is shown that the proposed ROM offers greater performance in prediction of
full-order states when compared to a popular method employing POD and GPR over
a number of ROM dimensions
