Determining the coupling between systems remains a topic of active research in the field of complex science. Identifying the proper causal influences in time series can already be very challenging in the trivariate case, particularly when the interactions are non-linear. In this paper, the coupling between three Lorenz systems is investigated with the help of specifically designed artificial neural networks, called time delay neural networks (TDNNs). TDNNs can learn from their previous inputs and are therefore well suited to extract the causal relationship between time series. The performances of the TDNNs tested have always been very positive, showing an excellent capability to identify the correct causal relationships in absence of significant noise. The first tests on the time localization of the mutual influences and the effects of Gaussian noise have also provided very encouraging results. Even if further assessments are necessary, the networks of the proposed architecture have the potential to be a good complement to the other techniques available in the market for the investigation of mutual influences between time series.

中文翻译：

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关于时延神经网络检测时间序列间间接耦合的潜力

确定系统之间的耦合仍然是复杂科学领域中活跃的研究课题。在三变量情况下，识别时间序列中的适当因果影响已经非常具有挑战性，尤其是当相互作用是非线性的时。在本文中，在专门设计的人工神经网络（称为时延神经网络 (TDNN)）的帮助下，研究了三个洛伦兹系统之间的耦合。TDNN 可以从它们之前的输入中学习，因此非常适合提取时间序列之间的因果关系。测试的 TDNN 的性能一直非常积极，显示出在没有显着噪声的情况下识别正确因果关系的出色能力。对相互影响和高斯噪声影响的时间定位的首次测试也提供了非常令人鼓舞的结果。即使需要进一步评估，所提议架构的网络也有可能成为市场上其他可用技术的良好补充，用于调查时间序列之间的相互影响。