Data-driven nonlinear aerodynamics models with certifiably optimal boundedness properties

Obtaining predictive low-order models is a central challenge in fluid dynamics. Data-driven frameworks have been widely used to obtain low-order models of aerodynamic systems; yet, resulting models tend to yield predictions that grow unbounded with time. Recently introduced stability-promoting methods can facilitate the identification of bounded models, but tend to require extensive brute-force tuning even in the context of simple academic systems. Here, we show how recent theoretical advances in the long-term boundedness of dynamical systems can be integrated into data-driven modelling frameworks to ensure that resulting models will yield bounded predictions of incompressible flows. Specifically, we propose to solve a particular set of convex semidefinite programming problems to (i) certify whether a system admits a globally attracting bounded set for the chosen modelling parameters, and (ii) compute a model with the optimal (tightest) bound on this globally attracting set. We demonstrate the approach via integration within the sparse identification of nonlinear dynamics (SINDy) modelling framework. Application on two low-order benchmark problems establishes the merits of the approach. We then apply our approach to obtain a low-order (six-mode) model of unsteady separation over a NACA-65(1)-412 aerofoil at ${\textit{Re}}=20\,000$ – a flow that has been notoriously difficult to model using data-driven methods. The resulting model is found to accurately predict the dynamics of unsteady separation, with model predictions remaining bounded indefinitely. We anticipate this work will benefit future efforts in modelling strongly nonlinear flows, especially in settings where physically viable long-term forecasts are paramount.