Physics as the Inductive Bias for Causal Discovery

Causal discovery is often a data-driven paradigm to analyze complex real-world systems. In parallel, physics-based models such as ordinary differential equations (ODEs) provide mechanistic structure for many dynamical processes. Integrating these paradigms potentially allows physical knowledge to act as an inductive bias, improving identifiability, stability, and robustness of causal discovery in dynamical systems. However, such integration remains challenging: real dynamical systems often exhibit feedback, cyclic interactions, and non-stationary data trend, while many widely used causal discovery methods are formulated under acyclicity or equilibrium-based assumptions. In this work, we propose an integrative causal discovery framework for dynamical systems that leverages partial physical knowledge as an inductive bias. Specifically, we model system evolution as a stochastic differential equation (SDE), where the drift term encodes known ODE dynamics and the diffusion term corresponds to unknown causal couplings beyond the prescribed physics. We develop a scalable sparsity-inducing MLE algorithm that exploits causal graph structure for efficient parameter estimation. Under mild conditions, we establish guarantees to recover the causal graph. Experiments on dynamical systems with diverse causal structures show that our approach improves causal graph recovery and produces more stable, physically consistent estimates than purely data-driven state-of-the-art baselines.

💡 Research Summary

This paper introduces a novel framework that leverages partial physical knowledge as an inductive bias for causal discovery in dynamical systems. The authors model system evolution with a stochastic differential equation (SDE) in which the drift term encodes known ordinary differential equation (ODE) dynamics, while the diffusion term captures unknown causal couplings beyond the prescribed physics. By treating the diffusion matrix as a sparse linear operator, they turn the problem of learning the causal graph into a sparsity‑inducing maximum‑likelihood estimation.

A scalable gradient‑based algorithm, SCD (Scalable Causal Discovery), is derived from a quasi‑likelihood obtained via Euler‑Maruyama discretization. SCD exploits the graph’s sparsity to achieve computational efficiency and incorporates an L1 regularizer that directly promotes a sparse adjacency matrix. The authors prove a high‑probability recovery guarantee: for each node i with s_i parents, if the number of samples n satisfies n ≳ max{s_i³ log p, s_i⁴} and the regularization λ_n is chosen on the order of (p log p + p s_i)/n, then SCD recovers the true causal graph with overwhelming probability. This result holds without assuming acyclicity, additive‑noise, or stationarity, distinguishing it from prior work.

Empirical evaluation spans synthetic SDEs with both acyclic and cyclic structures, as well as realistic scenarios such as coupled oscillators, robotic arm dynamics, and epidemic spread models. Across four state‑of‑the‑art baselines—including DYNOTEARS, VAR‑based Granger, Neural‑ODE‑based NGM, and variational SCOTCH—SCD consistently achieves higher F1 scores for graph recovery and yields more stable, physically consistent parameter estimates. Notably, the method remains effective when feedback loops are present, demonstrating that physics‑based inductive bias can alleviate the need for explicit acyclicity constraints.

The paper also discusses limitations, such as the linearity assumption on the diffusion term and the lack of a full Bayesian treatment of uncertainty, and outlines future directions including nonlinear diffusion modeling, Bayesian SDE inference, and application to real experimental data. Overall, the work provides a theoretically grounded and practically robust approach to integrate mechanistic physics with data‑driven causal discovery, opening new avenues for analyzing complex, non‑stationary, and feedback‑rich systems.