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

Abstract

Modeling realistic fluid and plasma flows is computationally intensive, motivating the use of reducedorder 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 [1] 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.

Machine Learning for Fluid Mechanics

“…The sparse identification of nonlinear dynamics algorithm [18] learns dynamical systems models with as few terms from a library of candidate terms as are needed to describe the training data. There are several formulations involving different loss terms and optimization algorithms that promote additional physical notions, such as stability [99] and energy conservation [100]. Stability promoting loss functions based on notions of Lyapunov stability have also been incorporated into autoencoders, with impressive results on fluid systems [101].…”

“…One approach is to explicitly add constraints to the optimization, for example that certain coefficients must be non-negative, or that other coefficients must satisfy a specified algebraic relationship with each other. Depending on the given machine learning architecture, it may be possible to enforce energy conservation [100] or stability constraints [99] in this way. Another approach involves employing custom optimization algorithms required to minimize the physically motivated loss functions above, which are often non-convex.…”

System Identification of Two-Dimensional Transonic Buffet

“…In this work, the simple hard-thresholded least-squares algorithm proposed in [47] was however found to be sufficient. Note finally that, since its introduction, numerous extensions of SINDy have been proposed, see for instance [56,[59][60][61][62][63][64][65][66][67][68] and references therein.…”

On closures for reduced order models $-$ A spectrum of first-principle to machine-learned avenues

“…Indeed, the results of integrating the model derived by SINDy outperformed those from standard GROM. A recent innovation 359 , trapping SINDy, identifies a dynamics with a trapping region, i.e., guarantees boundedness of the solution 114 . This innovation is particularly important for higher-dimensional dynamics, where sparse identification is prone to give rise to unbounded solutions otherwise.…”