Predictive POD-Galerkin Models for Streamfunction–Vorticity Equation with Data-Driven Closure

Galerkin reduced order models are computationally inexpensive representations of the full order governing equations, however when execution time is significantly decreased it usually comes at a cost of reduced accuracy due to mode truncation since their absence distorts the flow systems energy cascade and neglects other important dynamics, hence the system becomes unstable after a while if not compensated for. In this thesis the streamline vorticity equation and its corresponding Galerkin system is considered as a benchmark problem to study data-driven closure model which is able to predict the corrections for the modal coefficient in order for the Galerkin system to match its corresponding full dimensional model restricted to the subspace spanned by the POD modes. Successively increasing the number of POD modes in the Galerkin system from 40, 50 to 60 improves the accuracy of the Galerkin model and its data-driven closure model since the subspace is able to model more complex dynamics with increased number of modes hence the learned closure model is more accurate. This work also inspires how modern reduced order modeling approaches like generative AI as a starting point (in contrast to traditional ROMs as a starting point which is considered in this thesis) can be complemented with the benefits of traditional ROM analogs like POD manifold which has optimality and orthogonality benefits and Galerkin projection, for example by training a generative AI model to construct a nonlinear manifold, and evaluate it while ensuring that the residual norm of the governing equations is satisfied, allowing the model to predict outcomes for situations not present in the training set, in the spirit of a Galerkin ROM.