From Ordinary Differential Equations to Structural Causal Models: the deterministic case

We show how, and under which conditions, the equilibrium states of a first-order Ordinary Differential Equation (ODE) system can be described with a deterministic Structural Causal Model (SCM). Our exposition sheds more light on the concept of causality as expressed within the framework of Structural Causal Models, especially for cyclic models.

💡 Research Summary

The paper establishes a rigorous connection between deterministic first‑order ordinary differential equation (ODE) systems and deterministic structural causal models (SCMs). Starting from a set of variables (X_i) indexed by (i\in I={1,\dots,D}) and a dynamical system described by (\dot X_i(t)=f_i\bigl(X_{\text{pa}D(i)}\bigr)) with initial condition (X_i(0)=X{0,i}), the authors associate a directed graph (G_D) whose edges reflect the functional dependencies of the ODE.

A central contribution is the formal treatment of “perfect interventions” (the do‑operator) within this continuous‑time framework. For any subset (I\subseteq I) and target values (\xi_I), the intervention (do(X_I=\xi_I)) is modeled by adding a strong feedback term (\kappa(\xi_i-X_i)) to each intervened equation and letting (\kappa\to\infty). In the limit the dynamics of the intervened variables become (\dot X_i=0) with fixed initial condition (X_i(0)=\xi_i). Graphically, all incoming edges to the intervened nodes are removed, reflecting the causal cut introduced by the intervention.

Stability is required for the translation to be meaningful. An ODE is called stable if, for every initial state, the trajectory converges to a unique equilibrium (X^*) as (t\to\infty). The definition is extended to a set of interventions (\mathcal J): the system must converge to a unique equilibrium for each intervention in (\mathcal J). Under this strong stability assumption, the equilibrium equations of the observational system are simply (0=f_i\bigl(X_{\text{pa}_D(i)}\bigr)) for all (i). For an intervention (do(X_I=\xi_I)) the equilibrium equations become (0=X_i-\xi_i) for (i\in I) together with the unchanged equations for the remaining variables.

A key insight is that the set of equilibrium equations, taken as an unlabeled collection, loses the information needed to predict the effect of interventions because it no longer indicates which equation corresponds to which variable. The authors therefore introduce Labeled Equilibrium Equations (LEE): each equation is paired with its index ((i,E_i)). This labeling preserves the mapping from variables to equations, allowing a systematic replacement of the appropriate equations under any perfect intervention. Consequently, the LEE representation encodes the full causal semantics of the original ODE, including cyclic dependencies.

Two illustrative examples are provided. The Lotka‑Volterra predator‑prey model, which is inherently unstable and possesses two equilibria (the trivial zero state and a positive coexistence point), demonstrates how a perfect intervention on the predator population ((do(X_2=\xi_2)) ) typically yields a stable equilibrium ((0,\xi_2)). The second example concerns a damped mass‑spring system with friction. Each mass‑position‑momentum pair ((Q_i,P_i)) forms a variable; friction guarantees convergence to a unique equilibrium for any initial condition. Fixing the position of a subset of masses (intervention (do(Q_i=\xi_i,P_i=0))) still results in a unique equilibrium, illustrating stability with respect to a broad class of interventions.

The paper’s contributions can be summarized as follows:

1. Equivalence of ODE equilibria and deterministic SCMs – By labeling equilibrium equations, the authors show that any stable ODE (including cyclic ones) induces a well‑defined deterministic SCM that captures the causal effect of perfect interventions.
2. Causal interpretation of cycles – Cyclic causal graphs, traditionally problematic in the acyclic SCM literature, acquire a natural meaning as the equilibrium of an underlying dynamical system.
3. Intervention calculus without explicit time – The do‑operator is realized by substituting the appropriate labeled equilibrium equations, eliminating the need to model the full continuous‑time dynamics when predicting intervention outcomes.
4. Robustness across dynamics – Different ODE specifications can lead to the same labeled equilibrium system, implying that observational equilibrium data alone may not identify the underlying dynamics, only the induced SCM.

Overall, the work bridges dynamical systems theory and causal inference, providing a principled method to derive causal models from deterministic ODEs, to handle feedback loops, and to reason about interventions at equilibrium. This framework opens avenues for applying causal analysis in physics, biology, and engineering contexts where systems are naturally described by differential equations.