# Deep Koopman operator framework for causal discovery in nonlinear dynamical systems

arXiv: https://arxiv.org/abs/2505.14828 · HTML: https://mlsanity.com/paper/2505.14828/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 63 references that are in the map.
- First posted on arXiv 20 May 2025.

## Builds on (most cited first, 17 of 17)

- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
- [Discovering contemporaneous and lagged causal relations in autocorrelated nonlinear time series datasets](https://mlsanity.com/paper/2003.03685/) (2020)
- [Forecasting Sequential Data using Consistent Koopman Autoencoders](https://mlsanity.com/paper/2003.02236/) (2020)
- [On Convergence of Extended Dynamic Mode Decomposition to the Koopman Operator](https://mlsanity.com/paper/1703.04680/) (2017)
- [Learning Dynamical Systems via Koopman Operator Regression in Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/2205.14027/) (2022)
- [ChaosBench: A Multi-Channel, Physics-Based Benchmark for Subseasonal-to-Seasonal Climate Prediction](https://mlsanity.com/paper/2402.00712/) (2024)
- [Koopman Neural Forecaster for Time Series with Temporal Distribution Shifts](https://mlsanity.com/paper/2210.03675/) (2022)
- [Discovering Causal Relations and Equations from Data](https://mlsanity.com/paper/2305.13341/) (2023)
- [Causal models for dynamical systems](https://mlsanity.com/paper/2001.06208/) (2020)
- [From Deterministic ODEs to Dynamic Structural Causal Models](https://mlsanity.com/paper/1608.08028/) (2016)
- [Generative learning for nonlinear dynamics](https://mlsanity.com/paper/2311.04128/) (2023)
- [Dynamic Structural Causal Models](https://mlsanity.com/paper/2406.01161/) (2024)
- [Analyzing Koopman approaches to physics-informed machine learning for long-term sea-surface temperature forecasting](https://mlsanity.com/paper/2010.00399/) (2020)
- [Generative emulation of chaotic dynamics with coherent prior](https://mlsanity.com/paper/2504.14264/) (2025)
- [On Principles of Emergent Organization](https://mlsanity.com/paper/2311.13749/) (2023)
- [Tangent Space Causal Inference: Leveraging Vector Fields for Causal Discovery in Dynamical Systems](https://mlsanity.com/paper/2410.23499/) (2024)

## Built on by (most cited first, 2 of 2)

- [CausalDynamics: A large-scale benchmark for structural discovery of dynamical causal models](https://mlsanity.com/paper/2505.16620/) (2025)
- [Application of Reduced-Order Models for Temporal Multiscale Representations in the Prediction of Dynamical Systems](https://mlsanity.com/paper/2510.18925/) (2025)

## Abstract (arXiv)

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.