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From Ordinary Differential Equations to Structural Causal Models: the deterministic case
Joris M. Mooij, Dominik Janzing, Bernhard Schölkopf
TL;DR
Cyclic causal models can represent feedback, but existing relationships among causal-modeling approaches do not generally extend to that setting, leaving the interpretation of cyclic equations at issue. The paper derives deterministic SCMs from ODE equilibria and shows that their intervention semantics remain valid. The resulting equilibrium representation is useful when temporal resolution is too coarse for the equilibration timescale, though it omits convergence-rate information.
Problem
Cyclic models extend causal modeling to feedback, but the relationships among existing approaches do not generally extend to the cyclic case, and cyclic structural equations require interpretation.
Source: §1 · p. 1Method
The paper maps labeled equilibrium equations from deterministic ODE systems to deterministic SCMs, using structural solvability to define intervention response functions.
Source: §1 · p. 1, §4 · p. 6, §4.4 · p. 7Results
ODE equilibria yield time-independent SCMs that retain valid intervention semantics, and different dynamics can yield identical SCMs.
Source: §1 · p. 1Takeaways & Limitations
When equilibration is too fast for measurement resolution, an SCM may be more natural than a dynamical-system representation; snapshot or equilibrium data can also be represented by an LEE or SCM.
Source: §5 · p. 8Equilibrium representations lose information about the rate of convergence, so a dynamical-system representation may be more natural when time-series data are available.
Source: §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
Acyclic causal-model approaches are well understood, but their relationships do not generally extend to cyclic models, despite efforts to model feedback. This section presents an ODE-equilibrium interpretation of cyclic SCMs that preserves intervention semantics, while restricting the discussion to deterministic systems.
1 Introduction
- Cyclic causal models can represent feedback, but relationships among causal-modeling approaches do not generally carry over from the acyclic case.Partial results are known for linear and discrete cases, and algorithms for cyclic inference have been proposed from different assumptions.Source: §1 · p. 1
- ODE equilibria yield time-independent SCMs whose intervention semantics remain valid, without treating the structural equations as discrete-time update rules.The structural equations arise from the dynamics, and different dynamics can yield identical SCMs.Source: §1 · p. 1
- Time order alone cannot establish that an earlier variable caused a later one, since both may instead have a common cause.The paper therefore distinguishes causal semantics from simply inferring the time structure connecting observed variables.Source: §1 · pp. 1–2
- A causal graph can describe how interventions change other variables’ equilibrium states even when variables do not represent measurements at particular times.At this phenomenological level, the original time structure is lost while cyclic structural equations retain a well-defined meaning.Source: §1 · p. 2
- The discussion considers deterministic systems and leaves stochastic systems with possible confounding for future work.Source: §1 · p. 2
2 Ordinary Differential Equations
The section represents coupled first-order ODEs as dependency graphs and defines perfect interventions and stability conditions. Its examples show that equilibrium behavior depends on the dynamics and intervention: Lotka–Volterra is not stable observationally, while the damped mass-spring system retains unique equilibria under physically realizable position interventions.
2.1 Observational system
- An ODE specifies each variable’s time derivative as a function of its parent variables, and its graph has an edge when one variable’s value affects another’s derivative.The system is assumed to describe the natural, observational state.Source: §2.1 · p. 2
2.1.1 Example: the Lotka-Volterra model
- The Lotka–Volterra example models mutual influence between prey and predator abundances, with a graph encoding their dependencies.The variables can also be parents of themselves.Source: §2.1.1 · p. 2, Figure 1 · p. 2
2.2 Perfect interventions
- A perfect intervention fixes selected variables at constant target values from t = 0 onward, overriding their usual mechanisms and removing their incoming graph arrows.One ODE realization adds κ(ξi − Xi) feedback terms and takes κ → ∞, assuming the relevant functions are bounded.Source: §2.2 · pp. 2–3
2.2.1 Example: Lotka-Volterra model
- In the Lotka–Volterra example, do(X2 = ξ2) can be realized by holding predator abundance at its target value throughout the intervention.The corresponding graph is shown in Figure 1(b).Source: §2.2.1 · p. 3
2.3 Stability
- Stability requires a unique equilibrium reached from every initial state; stability with respect to interventions additionally requires this for each specified intervention and target.The exposition adopts these strong conditions for clarity, though the conditions can be weakened.Source: §2.3 · p. 3
2.3.1 Example: the Lotka-Volterra model
- The observational Lotka–Volterra system is not stable: one equilibrium is unstable, while the other has undamped oscillations around it.The intervened system is generically stable with unique equilibrium (0, ξ2) when θ11 − θ12ξ2 ≠ 0; equality instead gives equilibria (c, ξ2), c ≥ 0.Source: §2.3.1 · p. 3
2.3.2 Example: damped harmonic oscillators
- The damped mass-spring system has a unique equilibrium where forces balance, and friction makes each mass converge to it over time.Fixing one or more masses at other positions changes the remaining masses’ equilibrium positions.Source: §2.3.2 · pp. 3–4, Figure 2 · p. 3
- Fixing positions through interventions on the corresponding position–momentum pairs preserves convergence to a unique equilibrium for any number of constrained masses.Interventions specifying nonzero momentum while fixing the corresponding position are physically impossible and are excluded.Source: §2.3.2 · p. 4, Figure 3 · p. 4
3 Equilibrium equations
The section derives equilibrium equations from observational and intervened ODEs, then shows that labeling those equations preserves the information needed to represent equilibrium changes under perfect interventions.
3.1 Observational system
- At equilibrium, each variable’s rate of change is zero, and stability guarantees a unique solution to the resulting observational equations.The equations are coupled and have the system variables as unknowns.Source: §3.1 · p. 4
3.2 Intervened systems
- Under intervention, replace each targeted variable’s equilibrium equation with 0 = Xi − ξi, leaving the other equations unchanged.This replacement follows from the intervened ODE’s equilibrium equations.Source: §3.2 · pp. 4–5, §3.3 · p. 5
3.3 Labeling equilibrium equations
- Unlabeled equilibrium equations lose which variable each equation belongs to, so they cannot identify which equation to replace during an intervention.The ODE’s time-derivative terms identify intervention targets, but these become zero in the equilibrium equations.Source: §3.3 · p. 5
3.4 Labeled equilibrium equations
- A Labeled Equilibrium Equation system (LEE) associates each equation with its variable; its graph records dependencies, and solvability means a unique solution, including after specified interventions.A perfect intervention replaces the labeled equations for its target variables.Source: §3.4 · p. 5
- Applying a perfect intervention to an ODE’s induced LEE gives the same result as constructing the LEE from the intervened ODE; stability also makes their equilibria identical.The latter correspondence holds when the ODE is stable with respect to the intervention.Source: §3.4 · p. 5
3.5 Example: damped harmonic oscillators
- The damped, coupled harmonic oscillator example specifies its labeled equilibrium equations explicitly.Source: §3.5 · p. 6
4 Structural Causal Models
The paper constructs a deterministic SCM from labeled equilibrium equations and relates its solutions and interventions to those of an ODE. The correspondence holds under stated stability and solvability conditions, including for cyclic systems.
4.1 Observational; 4.2 Interventions; 4.3 Solvability
- An SCM equation assigns each variable a function of its parents, and perfect intervention replaces the equations for the forced variables.Acyclic deterministic SCMs are solvable for all intervention targets, but this need not hold when directed cycles are present.Source: §4.1 · p. 6, §4.2 · p. 6, §4.3 · p. 6
4.4 From labeled equilibrium equations to deterministic SCMs
- Structural solvability of an LEE requires solvability under selected interventions for each variable, including its non-self parents among the intervention targets.Source: §4.4 · p. 6
- For each variable, the construction uses interventions on its parent variables to define a function, then writes the SCM equation Xi = hi(XpaME(i)).The induced equation and its labeled equation have identical solution sets.Source: §4.4 · p. 7
- When both an LEE and its intervened version are structurally solvable, intervening on its induced SCM agrees with constructing an SCM from the intervened LEE, and the corresponding solutions match under the stated solvability condition.Source: §4.4 · p. 7
4.5 From ODEs to deterministic SCMs
- If the ODE and its intervened version are structurally stable, the mappings in Figure 4 commute; if the original ODE is also stable under that intervention, the SCM has the same unique solution as the intervened ODE’s stable equilibrium.The structural-stability conditions are defined through stability under selected interventions.Source: §4.5 · p. 7, Figure 4 · p. 8
- The induced SCM has no self-loops even when the ODE has them; in cyclic cases, solving the SCM still requires solving equations, much as with the LEE.The paper notes that structural stability implies the existence of self-loops in the ODE because otherwise an equilibrium variable would be undetermined.Source: §4.5 · pp. 7–8
- The LEE and SCM both retain the information needed to predict arbitrary perfect interventions, but conversion from an SCM back to an LEE is not unique.Source: §4.5 · p. 7
4.6 Example: damped harmonic oscillators
- For the damped harmonic oscillator example, the SCM uses positions alone because momentum variables vanish at equilibrium, and each mass’s position is directly influenced by its neighbors.Source: §4.6 · p. 8
5 Discussion
The discussion presents an explicit bridge from differential-equation models to causal models, while noting that equilibrium representations discard information and that the construction has unresolved scope limits. It also identifies stochastic systems with confounding as future work.
5 Discussion
- The paper builds an explicit bridge from differential equations, commonly used in empirical sciences and engineering, to causal models.Figure 5 depicts the induced SCM graph for the mass-spring system with D=4.Source: §5 · p. 8, Figure 5 · p. 8
- Equilibrium representations lose information, particularly the rate of convergence; time-series data may favor a dynamical-system representation, while snapshot or equilibrium data may suit an LEE or SCM.Source: §5 · p. 8
- The paper presents one way to derive SCMs, not the only way, and leaves constants of motion unresolved; unfixed oscillator endpoints can make total momentum depend on initial conditions and destabilize the dynamics under its definition.Source: §5 · p. 8
- The authors plan to extend the framework to stochastic systems, where uncertainty in equation parameters or latent variables introduces confounding as a complication.Source: §5 · pp. 8–9