Promoting global stability in data-driven models of quadratic nonlinear\n dynamics

Abstract

Modeling realistic fluid and plasma flows is computationally intensive,\nmotivating the use of reduced-order models for a variety of scientific and\nengineering tasks. However, it is challenging to characterize, much less\nguarantee, the global stability (i.e., long-time boundedness) of these models.\nThe seminal work of Schlegel and Noack (JFM, 2015) provided a theorem outlining\nnecessary and sufficient conditions to ensure global stability in systems with\nenergy-preserving, quadratic nonlinearities, with the goal of evaluating the\nstability of projection-based models. In this work, we incorporate this theorem\ninto modern data-driven models obtained via machine learning. First, we propose\nthat this theorem should be a standard diagnostic for the stability of\nprojection-based and data-driven models, examining the conditions under which\nit holds. Second, we illustrate how to modify the objective function in machine\nlearning algorithms to promote globally stable models, with implications for\nthe modeling of fluid and plasma flows. Specifically, we introduce a modified\n"trapping SINDy" algorithm based on the sparse identification of nonlinear\ndynamics (SINDy) method. This method enables the identification of models that,\nby construction, only produce bounded trajectories. The effectiveness and\naccuracy of this approach are demonstrated on a broad set of examples of\nvarying model complexity and physical origin, including the vortex shedding in\nthe wake of a circular cylinder.\n