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From Ordinary Differential Equations to Structural Causal Models: the deterministic case
Joris Mooij, Dominik Janzing, Bernhard Schoelkopf
TL;DR
The paper asks how causal models can represent feedback systems when time order alone does not settle causal relations. It derives deterministic SCMs from ODE equilibrium behavior and shows that, under stated stability conditions, the induced SCM’s solution coincides with the ODE’s stable equilibrium.
Problem
Time order cannot determine whether an earlier variable caused a later one or whether both have a common cause, while cyclic SCMs need a well-defined causal interpretation.
Source: §1 · pp. 1–2Method
The paper derives a deterministic SCM from a first-order ODE’s labeled equilibrium equations, preserving intervention targets.
Source: §1 · p. 1, §3.3 · p. 5, §3.4 · p. 5Results
When the ODE and intervened ODE are structurally stable, the induced SCM has a unique solution that, when the intervened ODE is stable, coincides with its stable equilibrium.
Source: §4.5 · p. 7Takeaways & Limitations
Equilibrium SCMs can represent intervention effects when equilibration is too fast for measurement resolution, while retaining less dynamic information than the ODE representation.
Source: §5 · p. 8The treatment is deterministic; equilibrium representations also lose information about rates of convergence, so time-series data may favor a dynamical-system representation.
Source: §2.1 · p. 2, §5 · p. 8
Abstract
from arXiv · showhide
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.
1 Introduction
The paper addresses how causal relationships can be understood when feedback makes causal models cyclic, while time order alone cannot distinguish direct causation from a common cause. It derives a time-independent SCM interpretation from ODE equilibrium behavior, with interventions retaining causal semantics, and focuses on deterministic systems.
1 Introduction
- Cyclic causal models have been studied, but relationships among causal-modeling approaches do not generally extend to the cyclic case.Partial results are known for linear and discrete cases, and algorithms have been proposed to infer cyclic models from observational data.Source: §1 · p. 1
- ODE equilibrium behavior yields a time-independent SCM whose intervention semantics remain valid, without treating structural equations as the continuous-time dynamics.Different dynamics can yield identical SCMs, and the interpretation does not substantially distinguish cyclic from acyclic cases.Source: §1 · p. 1
- Time order cannot establish whether an earlier variable caused a later one or whether both arise from a common cause.Causal graphs can instead describe how interventions influence equilibrium states, where the original time structure disappears.Source: §1 · pp. 1–2
- The exposition considers deterministic systems and leaves stochastic systems with possible confounding for future work.Source: §1 · p. 2
2 Ordinary Differential Equations
The section defines first-order ODE systems, perfect interventions, and stability, then illustrates their behavior with Lotka–Volterra populations and damped coupled oscillators.
2.1 Observational system
- A first-order ODE system specifies each variable’s derivative as a function of its parents and an initial state; its graph links variables to derivatives that depend on them.Source: §2.1 · p. 2
2.2 Intervened system
- A perfect intervention fixes selected variables at constant targets from t = 0 onward, overriding their usual mechanisms and removing their incoming graph arrows.One possible realization adds κ(ξi − Xi) feedback terms and takes κ → ∞ under suitable regularity conditions.Source: §2.2 · pp. 2–3
2.3 Stability
- Strong stability means every initial state converges to one unique equilibrium; intervention stability requires this for each allowed intervention and target.The exposition uses these strong conditions for clarity, while noting that they can be weakened.Source: §2.3 · p. 3
2.3.1 Example: the Lotka-Volterra model
- The Lotka–Volterra observational system is unstable: one equilibrium is unstable and the other supports undamped oscillations.After do(X2 = ξ2), a unique stable equilibrium (0, ξ2) exists generically when θ11−θ12ξ2 ≠ 0; at equality there is a family (c, ξ2), c ≥ 0.Source: §2.3.1 · p. 3
2.3.2 Example: damped harmonic oscillators
- Damped coupled masses converge to a unique equilibrium, and fixing positions changes the equilibrium positions of unconstrained masses.For interventions fixing any number of position-momentum pairs at (ξi, 0), the system still converges uniquely; fixing a position with nonzero momentum is physically excluded.Source: §2.3.2 · pp. 3–4
3 Equilibrium equations
The section derives equilibrium equations from an ODE and shows how labeling them preserves intervention targets. Under stability, the labeled representation supports the same intervention outcomes as the corresponding ODE.
3.1 Observational system
- At an observational equilibrium, every variable’s rate of change is zero, and stability ensures the resulting equilibrium equations have a unique solution.Source: §3.1 · p. 4
3.2 Intervened system
- A perfect intervention replaces each targeted equilibrium equation with 0 = Xi − ξi; stability with respect to that intervention ensures a unique solution.Source: §3.2 · pp. 4–5
3.3 Labeling equilibrium equations
- Labeling each equilibrium equation preserves which equation an intervention must replace, allowing the labeled set to encode equilibrium values and their changes under perfect interventions.Source: §3.3 · p. 5
3.4 Labeled equilibrium equations
- A Labeled Equilibrium Equation system assigns each variable a labeled equation with a parent set and function; interventions replace target equations, and solvability requires a unique solution.Source: §3.4 · p. 5
- Under stability of the intervened ODE, the induced intervened LEE is solvable and has the same equilibrium; intervention on the induced LEE agrees with inducing an LEE from the intervened ODE.Source: §3.4 · p. 5
3.4.1 Example: damped harmonic oscillators
- The damped, coupled harmonic oscillator example gives explicit labeled equilibrium equations.Source: §3.4.1 · p. 6
4 Structural Causal Models
The paper maps labeled equilibrium equations (LEEs) to deterministic Structural Causal Models (SCMs), then relates the induced SCM to the ODE under stability and solvability conditions. The construction preserves intervention results, including in cyclic settings.
4.1 Observational system
- A deterministic SCM assigns each variable an equation based on its parents, which are treated as its direct causes.The graph represents dependencies between variables and has no self-loops by definition.Source: §4.1 · p. 6
4.2 Intervened system
- A perfect intervention fixes selected variables and replaces their structural equations with equations setting them to the chosen values.The equation left-hand sides identify intervention targets.Source: §4.2 · p. 6
4.3 Solvability
- An SCM is solvable when its equations have a unique solution; acyclic SCMs are solvable for all intervention targets, but cyclic ones need not be.Solvability with respect to a target set requires unique solutions after every intervention on that set.Source: §4.3 · p. 6
4.4 From labeled equilibrium equations to deterministic SCMs
- If each LEE remains solvable when all variables except one are fixed, the induced SCM equations can be constructed from the resulting unique values.Each induced structural equation has the same solution set as its corresponding labeled equation.Source: §4.4 · pp. 6–7
- When the original and intervened LEEs are structurally solvable, intervening on the induced SCM matches constructing an SCM from the intervened LEE, with identical solutions.The correspondence holds even when intervention replaces equations in both representations.Source: §4.4 · p. 7
4.5 From ODEs to deterministic SCMs
- If the ODE and its intervened version are structurally stable, the mapping diagram commutes; with additional stability of the intervened ODE, the SCM solution equals its stable equilibrium.The paper notes that an LEE may be more natural for cyclic systems because it does not require the additional structural stability assumptions of the SCM construction.Source: §4.5 · pp. 7–8
- The induced SCM has no self-loops, even when the ODE has them; the paper notes that structural stability implies self-dependence in the equilibrium equations.This distinction concerns the representation, not whether the ODE’s time derivative depends on the variable itself.Source: §4.5 · p. 7
4.5.1 Example: damped harmonic oscillators
- For damped harmonic oscillators, the example retains positions as SCM variables because momenta vanish at equilibrium, including under the considered interventions.The induced graph represents each mass’s position as directly influencing its neighbors’ positions at equilibrium.Source: §4.5.1 · p. 8
5 Discussion
When equilibration is too fast for measurement resolution, structural causal models may represent systems more naturally than differential equations. The paper builds an explicit bridge between these representations while noting what equilibrium views omit and that the derivation is not unique.
5 Discussion
- The paper builds an explicit bridge from differential equations to causal models, motivated by cases where equilibration is too fast for measurement resolution.In such cases, structural causal models may be the more natural representation.Source: §5 · p. 8
- Equilibrium representations discard convergence-rate information; time-series data may favor dynamics, while snapshot or equilibrium data may favor LEE or SCM.Source: §5 · p. 8
- This is one possible route to deriving SCMs, which may also be obtained from stochastic differential equations (Sokol and Hansen, 2013).Related work includes Voortman et al. (2010) and Iwasaki and Simon (1994).Source: §5 · p. 8
- The planned stochastic extension would accommodate parameter uncertainty, uncertain initial conditions for constants of motion, and latent variables for confounding.Source: §5 · p. 8