Causal Discovery in Nonlinear Dynamical Systems using Koopman Operators

TL;DR

The paper tackles causal discovery in nonlinear dynamical systems by formulating causal mechanisms as time-dependent flows in observable spaces and proving their equivalence to flow-map causality. It introduces a data-driven, Koopman-operator framework built on RKHS observables and Dynamic Mode Decomposition (DMD), employing random Fourier features to enable scalable, multivariate causal analysis in high-dimensional systems. The methodology yields a concrete, testable measure of causal influence by comparing marginal and joint Koopman evolutions of effect components, with time-resolved forecasting to capture scale-dependent causal effects. Demonstrations on coupled Rössler oscillators and the Lorenz 96 model reveal consistent causal flows that align with perturbation-based information propagation, validating the approach as a principled, data-driven toolkit for nonlinear dynamical causality. The work advances theory and practice by enabling causal analysis across time scales, handling confounding to an extent, and offering a route to high-dimensional causal discovery in spatiotemporal systems.

Abstract

We present a theory of causality in dynamical systems using Koopman operators. Our theory is grounded on a rigorous definition of causal mechanism in dynamical systems given in terms of flow maps. In the Koopman framework, we prove that causal mechanisms manifest as particular flows of observables between function subspaces. While the flow map definition is a clear generalization of the standard definition of causal mechanism given in the structural causal model framework, the flow maps are complicated objects that are not tractable to work with in practice. By contrast, the equivalent Koopman definition lends itself to a straightforward data-driven algorithm that can quantify multivariate causal relations in high-dimensional nonlinear dynamical systems. The coupled Rossler system provides examples and demonstrations throughout our exposition. We also demonstrate the utility of our data-driven Koopman causality measure by identifying causal flow in the Lorenz 96 system. We show that the causal flow identified by our data-driven algorithm agrees with the information flow identified through a perturbation propagation experiment. Our work provides new theoretical insights into causality for nonlinear dynamical systems, as well as a new toolkit for data-driven causal analysis.

Figures (11)

• Figure 1: Some causal phenomenology of the coupled Rössler oscillator system using a counterfactual causality measure. Column (a) shows causality from an asymmetrical system with $\Omega_2 \mathbf{\longrightarrow} \Omega_1$ and $\Omega_1 \mathbf{\centernot\longrightarrow} \Omega_2$. Column (b) shows sample time series of each degree of freedom with increasing coupling constants $c_1$.
• Figure 2: Causal measure per time of coupled Rössler.
• Figure 3: Graphical depiction of a marginal conditional forecast for 'effect' component $\Omega_E$. Predicted values $\widetilde{\omega}_E(t_n)$ are fed back into the model, shown with solid black arrows, while values $\omega_E(t_n)$ from the test data are plugged into the additional dictionary functions $\Psi$, as shown with dotted blue arrows.
• Figure 4: Conditional forecasts for $\Omega_2 \mathbf{\longrightarrow} \Omega_1$, $\Omega_1 \mathbf{\centernot\longrightarrow} \Omega_2$ coupled Rössler system.
• Figure 5: Causal graph depicting $\omega_1$ as a confounder for components $\omega_2$ and $\omega_3$

Theorems & Definitions (26)

• Definition 2.1
• Example 1
• Theorem 2.1
• Definition 3.1
• Definition 3.2
• Example 2
• Definition 3.3
• Remark 1
• Remark 2
• Example 3