# Causal Modeling of Dynamical Systems

arXiv: https://arxiv.org/abs/1803.08784 · HTML: https://mlsanity.com/paper/1803.08784/

## Key facts

- Cited by 24 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Mar 2018.

## Built on by (most cited first, 24 of 24)

- [Causal Structure Learning: a Combinatorial Perspective](https://mlsanity.com/paper/2206.01152/) (2022)
- [Causal Discovery from Temporal Data: An Overview and New Perspectives](https://mlsanity.com/paper/2303.10112/) (2023)
- [Causal Modeling with Stationary Diffusions](https://mlsanity.com/paper/2310.17405/) (2023)
- [Signature Kernel Conditional Independence Tests in Causal Discovery for Stochastic Processes](https://mlsanity.com/paper/2402.18477/) (2024)
- [Dynamic Structural Causal Models](https://mlsanity.com/paper/2406.01161/) (2024)
- [Causal Optimal Transport of Abstractions](https://mlsanity.com/paper/2312.08107/) (2023)
- [Fine-Grained Causal Dynamics Learning with Quantization for Improving Robustness in Reinforcement Learning](https://mlsanity.com/paper/2406.03234/) (2024)
- [Towards a Grounded Theory of Causation for Embodied AI](https://mlsanity.com/paper/2206.13973/) (2022)
- [A Measure-Theoretic Axiomatisation of Causality](https://mlsanity.com/paper/2305.17139/) (2023)
- [CausalDynamics: A large-scale benchmark for structural discovery of dynamical causal models](https://mlsanity.com/paper/2505.16620/) (2025)
- [Phenomenological Causality](https://mlsanity.com/paper/2211.09024/) (2022)
- [Causality and independence in perfectly adapted dynamical systems](https://mlsanity.com/paper/2101.11885/) (2021)
- [Curious Causality-Seeking Agents Learn Meta Causal World](https://mlsanity.com/paper/2506.23068/) (2025)
- [Aligning Perception, Reasoning, Modeling and Interaction: A Survey on Physical AI](https://mlsanity.com/paper/2510.04978/) (2025)
- [Causal Representation Meets Stochastic Modeling under Generic Geometry](https://mlsanity.com/paper/2602.05033/) (2026)
- [From Causal Discovery to Dynamic Causal Inference in Neural Time Series](https://mlsanity.com/paper/2603.20980/) (2026)
- [Do Not Marginalize Mechanisms, Rather Consolidate!](https://mlsanity.com/paper/2310.08377/) (2023)
- [When Counterfactual Reasoning Fails: Chaos and Real-World Complexity](https://mlsanity.com/paper/2503.23820/) (2025)
- [Adjustment formulas for learning causal steady-state models from closed-loop operational data](https://mlsanity.com/paper/2211.05613/) (2022)
- [Projecting infinite time series graphs to finite marginal graphs using number theory](https://mlsanity.com/paper/2310.05526/) (2023)
- [A Dynamical View of the Question of Why](https://mlsanity.com/paper/2402.10240/) (2024)
- [Observable Neural ODEs for Identifiable Causal Forecasting in Continuous Time](https://mlsanity.com/paper/2604.26070/) (2026)
- [Beyond Directed Acyclic Graphs: Causal Zeros and Causal Differential Equations](https://mlsanity.com/paper/2607.22910/) (2026)
- [Orca: Neural Operators for Causal Reasoning in Continuous Time](https://mlsanity.com/paper/2607.27867/) (2026)

## Abstract (arXiv)

Dynamical systems are widely used in science and engineering to model systems consisting of several interacting components. Often, they can be given a causal interpretation in the sense that they not only model the evolution of the states of the system's components over time, but also describe how their evolution is affected by external interventions on the system that perturb the dynamics. We introduce the formal framework of structural dynamical causal models (SDCMs) that explicates the causal semantics of the system's components as part of the model. SDCMs represent a dynamical system as a collection of stochastic processes and specify the basic causal mechanisms that govern the dynamics of each component as a structured system of random differential equations of arbitrary order. SDCMs extend the versatile causal modeling framework of structural causal models (SCMs), also known as structural equation models (SEMs), by explicitly allowing for time-dependence. An SDCM can be thought of as the stochastic-process version of an SCM, where the static random variables of the SCM are replaced by dynamic stochastic processes and their derivatives. We provide the foundations for a theory of SDCMs, by (i) formally defining SDCMs, their solutions, stochastic interventions, and a graphical representation; (ii) studying existence and uniqueness of the solutions for given initial conditions; (iii) providing Markov properties for SDCMs with initial conditions; (iv) discussing under which conditions SDCMs equilibrate to SCMs as time tends to infinity; (v) relating the properties of the SDCM to those of the equilibrium SCM. This correspondence enables one to leverage the wealth of statistical tools and discovery methods available for SCMs when studying the causal semantics of a large class of stochastic dynamical systems. The theory is illustrated with examples from different scientific domains.