Toward guaranteed stability in low-order data-driven models with closure modeling*

This work introduces a closure model that can ultimately be used to guarantee stability in  low-order, data-driven models. Previous work on developing closure models, such as projection onto quadratic or higher-order manifolds, do not typically guarantee stability. In this work we propose a trigonometric closure model with window functions that ensure the rate of change of energy is zero outside a finite region of the state-space.  This ensures that this closure model does not destabilize an already stable low-order model, but only serves to improve predictive fidelity.  Hyperparameters in the closure model are computed by minimizing the error between the truncated state variables and the closure terms over the training data. We demonstrate the closure modeling approach in conjunction with dynamic mode decomposition on several test cases, including a transient cylinder wake, a periodic cylinder wake, and a reduced-order model of plane Couette flow. Results show that the resulting models are always stable and exhibit improved predictive performance compared to models without the closure terms.