Causal PDE-Control Models (CPCMs)

• Causal PDE-Control Models (CPCMs) are dynamic frameworks that extend structural causal models to continuous-time systems using differential equations.
• They model explicit interventions by altering differential equations, capturing both deterministic and stochastic causal mechanisms along entire trajectories.
• CPCMs enable trajectory-level causal inference, providing actionable insights in fields such as ecology, biochemistry, and economics.

Causal PDE-Control Models (CPCMs) generalize structural causal models to continuous-time dynamical systems governed by partial (or ordinary/stochastic) differential equations, emphasizing the explicit modeling of interventions and causal mechanisms in the system’s evolution. CPCMs provide a rigorous mathematical and inferential framework for studying and controlling the causal effect of interventions on dynamic processes, enabling identification and quantification of cause-effect relationships over the entire trajectory rather than merely the stationary distribution.

1. Foundations: Causal Kinetic Models for Dynamical Systems

CPCMs originate from the extension of structural causal models (SCMs) to dynamical systems described by ODEs or SDEs. Two main classes are considered:

• Causal Kinetic Models with Measurement Noise: The deterministic system is governed by a set of ODEs with initial conditions, and randomness arises solely from measurement noise:
  • System: $x^k_t = f^k(x_t^{PA(k)})$, $x^k_0 = \xi^k_0$ for $k = 1,...,d$
  • Observation: $X^k_t = x^k_t + \epsilon^k_t$ with $\epsilon^k_t$ typically i.i.d.

The causal structure is encoded by the ODEs and initial conditions, and interventions affect the deterministic latent dynamics; noise is considered only in the observed process.

• Causal Kinetic Models with Driving Noise: Here, the SDE includes intrinsic stochasticity (driving noise) in the system evolution:
  • $$dX^k_t = f^k(X_t^{PA(k)})dt + h^k(X_t^{PA(k)})dW^k_t$$, $X^k_0 = \xi^k_0$

The noise terms $dW^k_t$ (Brownian motion increments) enter directly in the evolution, so randomness is a core part of the dynamic mechanism.

The distinction is critical: in the former, interventions reveal the impact against a deterministic backdrop, while in the latter, interventions may also modify or interact with the structure by which noise drives the system’s evolution.

2. Interventions and Causal Modularity in Differential Equations

In CPCMs, interventions are defined as explicit structural modifications of the generative (differential) assignments:

• Dynamic Interventions: Replace the structural equation for a variable. For example, a node $k$ originally governed by $\frac{dx^k_t}{dt} = f^k(x_t^{PA(k)})$ is redefined post-intervention as $\frac{dx^k_t}{dt} = g(x_t^{PA'(k)})$ for some new mechanism $g$ and altered parent set $PA'(k)$.
• Initial Condition Interventions: Set the initial condition, e.g., $do(x^k_0 := \xi)$, forcing the initial state of $x^k$.
• Stochastic Interventions: In SDE models, interventions may alter both drift and diffusion, as in $do(dX^k_t := g(X_t^{PA'})dt + j(X_t^{PA'})dW^k_t)$.

The principle of modularity carries over: only targeted assignments are modified, with all other causal relations invariant. This enables systematic “what if” analysis and supports causal effect identification by ensuring that changes propagate through the dynamic system consistently with the model’s structure.

3. Mathematical Structure and Trajectory-Based Causal Inference

CPCMs are formulated as continuous–time systems, with causality articulated in terms of the system’s response to interventions over entire solution trajectories, not just at equilibrium. Fundamental expressions are:

• Deterministic model (with measurement noise):
  • $x^k_t = f^k(x_t^{PA(k)})$; $X^k_t = x^k_t + \epsilon^k_t$
• Stochastic model (driving noise):
  • $$dX^k_t = f^k(X_t^{PA(k)})dt + h^k(X_t^{PA(k)})dW^k_t$$

Interventions are formalized as substitutions in the relevant structural differential assignment or initial condition. Analysis is thus driven by examining the post-intervention trajectory of the system, distinguishing CPCMs from models focused merely on stationary or equilibrium behavior.

This focus allows CPCMs to model fine-grained temporal responses, enabling richer questions about dynamic causal effects (e.g., time-dependent treatment effects, transient responses).

4. Applications and Canonical Examples

The framework applies across a spectrum of natural and engineered systems. Canonical examples include:

System	Equation(s) and Intervention Example	Domain(s)
Lotka–Volterra	$\frac{d[A]}{dt}=k_1[A]-k_2[A][B]$ $\frac{d[B]}{dt}=k_2[A][B]-k_3[B]$; Intervening by altering $k_1$ or initial $[A]$	Ecology, Chemistry
Michaelis–Menten	$\frac{d[P]}{dt} = c_1 [S]/(c_2+[S])$; Intervening by changing enzyme concentration or $c_1$	Biochemistry

Such applications illustrate that CPCMs can probe causal structure, perform interventional analysis, and explore hypothetical scenarios in real-world dynamical systems across biology, chemistry, and economics.

5. Challenges in Inference, Discovery, and Identifiability

CPCMs pose significant theoretical and practical challenges:

• Computational Complexity: Solving ODEs/SDEs post-intervention is substantially more demanding than in static SCMs, especially with nonlinearity or high dimensions.
• Parameter Identification: Estimation under dynamic noise (especially driving noise) requires adaptation of methods like gradient matching and nonlinear least squares to temporal settings.
• Causal Discovery: Standard conditional independence tests are inadequate due to temporal dependencies; alternative criteria such as local independence need to be developed and exploited.
• Noise and Errors-in-Variables: Deciding whether noise should be considered measurement or driving can significantly alter the identifiability of causal structure; incorrect assignment leads to biased inference.
• Hidden Variables and Equivalence: Unobserved states introduce additional inferential barriers, and dynamic systems may display complex equivalence classes beyond those familiar from static SCMs.
• Exploiting Invariance: While model modularity suggests that invariance principles can aid generalization across interventions, practical methods for leveraging invariance in dynamic settings remain an open research area.

These challenges define current research frontiers in the causal analysis of dynamical and PDE-governed systems.

6. Significance and Outlook

CPCMs extend the reach of causal inference to continuous–time systems, providing a formal schema for intervention and identification in a manner consistent with both the structure and dynamics of the underlying process. By modeling interventions at the level of the differential generator, CPCMs support nuanced trajectory-level causal inference—enabling, for example, precise predictions of interventions’ long-term or transient effects in complex adaptive systems.

Although such models can dramatically enrich the scope of causal analysis for time-dependent and non-stationary phenomena, their full implementation—especially for systems of PDEs—remains computationally and mathematically demanding. Open questions include optimal methods for parameter and structure learning, handling of noise (observation vs. driving), and principled exploitation of dynamic invariance and equivalence.

In summary, CPCMs provide a rigorous and generalizable framework for analyzing and controlling causal mechanisms in continuous–time, dynamic contexts. By integrating differential equations, structural interventions, and trajectory–based identification, they lay foundational ground for the causal analysis of complex temporal systems, opening new directions for research in dynamic causal discovery and control (Peters et al., 2020).