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

TL;DR

This paper integrates a theorem for global stability into data-driven fluid and plasma flow models, proposing modifications to machine learning objectives to ensure bounded, physically realistic long-term behavior.

Contribution

It introduces a stability-promoting modification to the SINDy algorithm, enabling the creation of models with guaranteed bounded trajectories for complex fluid and plasma dynamics.

Findings

Modified SINDy produces globally stable models

The approach is effective across diverse physical systems

Models maintain accuracy while ensuring stability

Abstract

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