Data-driven discovery of invariant measures

IF 2.9 3区 综合性期刊 Q1 MULTIDISCIPLINARY SCIENCES Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences Pub Date : 2024-03-27 DOI:10.1098/rspa.2023.0627
Jason J. Bramburger, Giovanni Fantuzzi
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Abstract

Invariant measures encode the long-time behaviour of a dynamical system. In this work, we propose an optimization-based method to discover invariant measures directly from data gathered from a system. Our method does not require an explicit model for the dynamics and allows one to target specific invariant measures, such as physical and ergodic measures. Moreover, it applies to both deterministic and stochastic dynamics in either continuous or discrete time. We provide convergence results and illustrate the performance of our method on data from the logistic map and a stochastic double-well system, for which invariant measures can be found by other means. We then use our method to approximate the physical measure of the chaotic attractor of the Rössler system, and we extract unstable periodic orbits embedded in this attractor by identifying discrete-time periodic points of a suitably defined Poincaré map. This final example is truly data-driven and shows that our method can significantly outperform previous approaches based on model identification.

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数据驱动的不变量测量发现
不变度量编码了动态系统的长期行为。在这项工作中,我们提出了一种基于优化的方法,直接从系统收集的数据中发现不变度量。我们的方法不需要明确的动力学模型,而且可以针对特定的不变度量,如物理和遍历度量。此外,它还适用于连续或离散时间的确定性和随机动力学。我们提供了收敛结果,并说明了我们的方法在逻辑图和随机双井系统数据上的性能,这些数据可以通过其他方法找到不变度量。然后,我们使用我们的方法来近似测量罗斯勒系统混乱吸引子的物理量,并通过识别适当定义的波恩卡莱图的离散时间周期点来提取嵌入该吸引子的不稳定周期轨道。最后这个例子是真正由数据驱动的,表明我们的方法大大优于以往基于模型识别的方法。
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来源期刊
CiteScore
6.40
自引率
5.70%
发文量
227
审稿时长
3.0 months
期刊介绍: Proceedings A has an illustrious history of publishing pioneering and influential research articles across the entire range of the physical and mathematical sciences. These have included Maxwell"s electromagnetic theory, the Braggs" first account of X-ray crystallography, Dirac"s relativistic theory of the electron, and Watson and Crick"s detailed description of the structure of DNA.
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