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

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.

对真实的流体和等离子体流动进行建模在计算上是密集型的，这促使在各种科学和工程任务中使用降阶模型。然而，对这些模型的全局稳定性（即长时间有界性）进行表征都具有挑战性，更不用说保证了。施莱格尔和诺阿克（《流体力学杂志》，2015年）的开创性工作提供了一个定理，概述了在具有能量守恒的二次非线性系统中确保全局稳定性的充分必要条件，目的是评估基于投影的模型的稳定性。在这项工作中，我们将该定理纳入通过机器学习获得的现代数据驱动模型中。首先，我们提出该定理应该成为基于投影和数据驱动模型稳定性的标准诊断方法，检查其成立的条件。其次，我们说明如何修改机器学习算法中的目标函数以促进全局稳定的模型，这对流体和等离子体流动的建模具有重要意义。具体来说，我们引入了一种基于非线性动力学稀疏识别（SINDy）方法的改进的“捕获SINDy”算法。这种方法能够识别出从构造上就只产生有界轨迹的模型。这种方法的有效性和准确性在一系列不同模型复杂度和物理起源的例子中得到了证明，包括圆柱尾流中的涡旋脱落。