Building dynamical stability into data-driven quadratic reduced-order models*

Quadratically nonlinear reduced-order models (ROMs) are commonly used for approximating the dynamics of fluids, plasmas, and many other physical systems. However, it is challenging to a-priori guarantee the local or global dynamical stability of reduced-order models built from data. For instance, a minimal requirement for physically-motivated ROMs is long-time boundedness for any initial condition, yet many ROMs in the literature still fail this basic requirement. For quadratically nonlinear systems with energy-preserving nonlinearities, the Schlegel and Noack trapping theorem (Schlegel and Noack 2015) provides necessary and sufficient conditions for long-time boundedness to hold. This analytic theorem was subsequently incorporated into system identification and machine learning techniques in order to  produce a-priori bounded models directly from data (Goyal 2023, Kaptanoglu 2021, Ouala 2023).

However, many dynamical systems exhibit weak breaking of the quadratically energy-preserving nonlinear structure required for the trapping theorem. To address this important case, we present recent work that relaxes the quadratically energy-preserving constraint and derives local stability guarantees for data-driven models. The analytic results are subsequently used with system identification techniques to build models with a-priori local stability properties (Peng 2024).

Lastly, we comment on alternative methods and future work for promoting dynamical stability in data-driven models.