  In this article, we introduce a modular hybrid analysis and modeling (HAM)
approach to account for hidden physics in reduced order modeling (ROM) of
parameterized systems relevant to fluid dynamics. The hybrid ROM framework is
based on using the first principles to model the known physics in conjunction
with utilizing the data-driven machine learning tools to model remaining
residual that is hidden in data. This framework employs proper orthogonal
decomposition as a compression tool to construct orthonormal bases and Galerkin
projection (GP) as a model to built the dynamical core of the system. Our
proposed methodology hence compensates structural or epistemic uncertainties in
models and utilizes the observed data snapshots to compute true modal
coefficients spanned by these bases. The GP model is then corrected at every
time step with a data-driven rectification using a long short-term memory
(LSTM) neural network architecture to incorporate hidden physics. A
Grassmannian manifold approach is also adapted for interpolating basis
functions to unseen parametric conditions. The control parameter governing the
system's behavior is thus implicitly considered through true modal coefficients
as input features to the LSTM network. The effectiveness of the HAM approach is
discussed through illustrative examples that are generated synthetically to
take hidden physics into account. Our approach thus provides insights
addressing a fundamental limitation of the physics-based models when the
governing equations are incomplete to represent underlying physical processes.
