Deep Koopman operator framework for causal discovery in nonlinear dynamical systems

2025-05-20

AI Summary

Quantifying causality in nonlinear dynamical systems remains challenging due to strong nonlinearity, multiscale coupling, and nonstationarity. To address this, we propose Kausal—a novel algorithm that integrates deep Koopman operator learning with reproducing kernel Hilbert space (RKHS) distance metrics for causal inference. Unlike conventional approaches, Kausal requires no predefined observables; instead, it automatically learns optimal observable functions and enables both causal direction identification and strength quantification within the Koopman linearization framework. The method combines theoretical interpretability with data-driven adaptability. Extensive evaluations on diverse nonlinear benchmark systems demonstrate significant improvements over classical methods such as Granger causality. Furthermore, Kausal successfully uncovers the dominant causal pathway and time-lagged mechanisms between sea surface temperature and wind fields in El Niño–Southern Oscillation (ENSO) events, establishing a new paradigm for attribution analysis in complex climate systems.

Technology Category

• Machine Learning: Causal Learning
• Reasoning under Uncertainty: Causality
• Knowledge Representation and Reasoning: Action, Change, and Causality

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Responsible Web: Algorithmic accountability and transparency on the web

Abstract

We use a deep Koopman operator-theoretic formalism to develop a novel causal discovery algorithm, Kausal. Causal discovery aims to identify cause-effect mechanisms for better scientific understanding, explainable decision-making, and more accurate modeling. Standard statistical frameworks, such as Granger causality, lack the ability to quantify causal relationships in nonlinear dynamics due to the presence of complex feedback mechanisms, timescale mixing, and nonstationarity. This presents a challenge in studying many real-world systems, such as the Earth's climate. Meanwhile, Koopman operator methods have emerged as a promising tool for approximating nonlinear dynamics in a linear space of observables. In Kausal, we propose to leverage this powerful idea for causal analysis where optimal observables are inferred using deep learning. Causal estimates are then evaluated in a reproducing kernel Hilbert space, and defined as the distance between the marginal dynamics of the effect and the joint dynamics of the cause-effect observables. Our numerical experiments demonstrate Kausal's superior ability in discovering and characterizing causal signals compared to existing approaches of prescribed observables. Lastly, we extend our analysis to observations of El Ni~no-Southern Oscillation highlighting our algorithm's applicability to real-world phenomena. Our code is available at https://github.com/juannat7/kausal.

Problem

• Identify cause-effect mechanisms in nonlinear dynamical systems
• Overcome limitations of Granger causality in nonlinear dynamics
• Apply deep Koopman operator for accurate causal discovery

Innovation

• Deep Koopman operator for causal discovery
• Deep learning infers optimal observables
• Causal estimates in kernel Hilbert space