NeuroKoopman Dynamic Causal Discovery

TL;DR

The paper addresses nonlinear Granger causality in multivariate time series by proposing NKDCD, a Koopman-inspired autoencoder that learns data-driven lifting φ and a linear VAR in the lifted space. It jointly optimizes the lifting, lifted-domain lag matrices $\{W_l\}$, and the projection φ^{-1} in an end-to-end fashion while enforcing structured group sparsity to reveal sparse causal dependencies. Validation across finance, Lorenz-96, fMRI, and DREAM3 datasets shows NKDCD often outperforms state-of-the-art nonlinear GC methods and linear VAR baselines, demonstrating robust recovery of both causal links and nonlinear dynamics. The approach has broad practical impact for domains such as neuroscience, economics, and systems biology, where understanding directed interactions in complex nonlinear systems is essential.

Abstract

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.

Figures (9)

• Figure 1: Proposed NKDCD Architecture
• Figure 2: VAR(3) data
• Figure 3: Estimated 5-step lag matrices
• Figure 4: Finance data
• Figure 5: Lorenz-96 model data

Theorems & Definitions (1)

• Definition 1: NKDCD Model-based GC