Local stability guarantees for data-driven quadratically nonlinear models

Abstract

Navier Stokes equations (NSEs) are complicated partial differential equations (PDEs) to describe the motion of fluids which are computationally expensive to simulate because of the high dimensionality. Reduced-order models (ROMs) are simpler models for evolving the flows by capturing only the dominant behaviors of a system. However it is challenging to guarantee the stability of these models either globally or locally. For quadratically nonlinear systems that represent many fluid flows, there is a theorem about global stability, which can be used to check if such ROMs are globally stable. Next, it was incorporated into a method that determine models directly from data, the sparse identification of nonlinear dynamics (SINDy) method in a modified technique called "trapping SINDy". In this work, we relax the quadratically energy-preserving constraints and promote local stability in data-driven models of quadratically nonlinear dynamics. This is important because this weakly quadratically energy-preserving structure exists in a large number of boundary conditions of fluids. First we raise a theorem outlining the sufficient condition to ensure local stability in linear-quadratic systems and provide an estimate stability radius based on the theorem. Second, we incorporate this theorem into data-driven models and present how we form the optimization problem based on the theorem. Then a modified "extended trapping SINDy" algorithm is introduced based on "trapping SINDy", enabling the trajectories of the data-driven models obtained via this method to be locally bounded inside a given ball-shape trapping region. Several examples are presented to demonstrate the effectiveness and accuracy of the proposed algorithm.