In many real-world applications where the system dynamics has an underlying interdependency among its variables (such as power grid, economics, neuroscience, omics networks, environmental ecosystems, and others), one is often interested in knowing whether the past values of one time series influences the future of another, known as Granger causality, and the associated underlying dynamics. This paper introduces a Koopman-inspired framework that leverages neural networks for data-driven learning of the Koopman bases, termed NeuroKoopman Dynamic Causal Discovery (NKDCD), for reliably inferring the Granger causality along with the underlying nonlinear dynamics. NKDCD employs an autoencoder architecture that lifts the nonlinear dynamics to a higher dimension using data-learned bases, where the lifted time series can be reliably modeled linearly. The lifting function, the linear Granger causality lag matrices, and the projection function (from lifted space to base space) are all represented as multilayer perceptrons and are all learned simultaneously in one go. NKDCD also utilizes sparsity-inducing penalties on the weights of the lag matrices, encouraging the model to select only the needed causal dependencies within the data. Through extensive testing on practically applicable datasets, it is shown that the NKDCD outperforms the existing nonlinear Granger causality discovery approaches.

翻译：在许多实际应用中（如电网、经济学、神经科学、组学网络、环境生态系统等），系统动态变量间存在潜在相互依赖关系。研究者通常关注一个时间序列的过去值是否影响另一个时间序列的未来值（即格兰杰因果关系），以及其关联的底层动态机制。本文提出一种基于Koopman理论的框架，利用神经网络对Koopman基进行数据驱动学习，称为NeuroKoopman动态因果发现（NKDCD），用于可靠推断格兰杰因果关系及其底层非线性动态。NKDCD采用自编码器架构，通过数据学习的基函数将非线性动态提升至高维空间，在该空间中提升后的时间序列可被可靠地线性建模。提升函数、线性格兰杰因果滞后矩阵以及投影函数（从提升空间到基空间）均以多层感知机表示，并同步联合学习。NKDCD还在滞后矩阵权重上施加稀疏性惩罚，促使模型仅选择数据中必要的因果依赖关系。通过在实用数据集上的广泛测试表明，NKDCD优于现有非线性格兰杰因果发现方法。