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 investigates a rigorous bridge between deterministic first‑order ordinary differential equation (ODE) systems and structural causal models (SCMs). Its central claim is that, under appropriate stability and uniqueness conditions, the equilibrium (steady‑state) of an ODE can be represented exactly by a deterministic SCM, and that interventions on the ODE correspond one‑to‑one with do‑operations on the SCM. The authors begin by formalising the ODE setting: a vector field (f:\mathbb{R}^n\to\mathbb{R}^n) defines the dynamics (\dot{x}=f(x)). They assume the existence of at least one equilibrium point (x^) satisfying (f(x^)=0). By invoking the Jacobian (J_f(x^)) and requiring all eigenvalues to have strictly negative real parts, they guarantee that the equilibrium is globally asymptotically stable, i.e., every trajectory converges to (x^). This stability is crucial because it ensures that the long‑run behavior of the system is independent of initial conditions, allowing a static description.

The next step is to translate the equilibrium condition into the language of SCMs. For each component (i), the authors define a parent set (\text{PA}_i) consisting of those variables that appear in the (i)-th component of the vector field (f_i). Solving the scalar equation (f_i(x^)=0) for (x_i^) yields a functional relationship \