Data-driven recovery of hidden physics in reduced order modeling of fluid flows

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 first principles to model the known physics in conjunction with utilizing the data-driven machine learning tools to model the remaining residual that is hidden in data. This framework employs proper orthogonal decomposition as a compression tool to construct orthonormal bases and a Galerkin projection (GP) as a model to build 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 Grassmann manifold approach is also adopted 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 then 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.