Quade Markus, Abel Markus, Nathan Kutz J, Brunton Steven L
Institut für Physik und Astronomie, Universität Potsdam, Karl-Liebknecht-Straße 24/25, 14476 Potsdam, Germany.
Department of Applied Mathematics, University of Washington, Seattle, Washington 98195, USA.
Chaos. 2018 Jun;28(6):063116. doi: 10.1063/1.5027470.
Big data have become a critically enabling component of emerging mathematical methods aimed at the automated discovery of dynamical systems, where first principles modeling may be intractable. However, in many engineering systems, abrupt changes must be rapidly characterized based on limited, incomplete, and noisy data. Many leading automated learning techniques rely on unrealistically large data sets, and it is unclear how to leverage prior knowledge effectively to re-identify a model after an abrupt change. In this work, we propose a conceptual framework to recover parsimonious models of a system in response to abrupt changes in the low-data limit. First, the abrupt change is detected by comparing the estimated Lyapunov time of the data with the model prediction. Next, we apply the sparse identification of nonlinear dynamics (SINDy) regression to update a previously identified model with the fewest changes, either by addition, deletion, or modification of existing model terms. We demonstrate this sparse model recovery on several examples for abrupt system change detection in periodic and chaotic dynamical systems. Our examples show that sparse updates to a previously identified model perform better with less data, have lower runtime complexity, and are less sensitive to noise than identifying an entirely new model. The proposed abrupt-SINDy architecture provides a new paradigm for the rapid and efficient recovery of a system model after abrupt changes.
大数据已成为新兴数学方法的关键促成要素，这些方法旨在自动发现动力系统，而在此类系统中，基于第一原理的建模可能难以处理。然而，在许多工程系统中，必须基于有限、不完整且有噪声的数据快速表征突变。许多领先的自动学习技术依赖于不切实际的大数据集，并且尚不清楚如何在突变后有效利用先验知识重新识别模型。在这项工作中，我们提出了一个概念框架，用于在低数据限制下应对突变时恢复系统的简约模型。首先，通过将数据的估计李雅普诺夫时间与模型预测进行比较来检测突变。接下来，我们应用非线性动力学的稀疏识别（SINDy）回归，通过添加、删除或修改现有模型项，以最少的变化更新先前识别的模型。我们在几个周期性和混沌动力系统中突变系统变化检测的示例上展示了这种稀疏模型恢复。我们的示例表明，与识别全新模型相比，对先前识别的模型进行稀疏更新在数据较少的情况下表现更好，运行时复杂度更低，并且对噪声不太敏感。所提出的突变-SINDy架构为突变后系统模型的快速有效恢复提供了一种新范式。
