Physics as the Inductive Bias for Causal Discovery
Jianhong Chen, Naichen Shi, Xubo Yue
TL;DR
This work addresses causal discovery in dynamical systems by embedding partial physical knowledge into a stochastic differential equation framework, where the drift g(t,X_t,γ) encodes known physics and the diffusion S_A(X_t) captures unknown causal couplings. It introduces SCD, a scalable gradient-based estimator that learns the causal graph from trajectory data via an Euler–Maruyama Gaussian quasi-likelihood with an ℓ1 penalty to promote sparsity; theory shows high-probability recovery of the true graph under mild Lyapunov-stability conditions, without requiring acyclicity or additive noise. Empirical results on both DAGs and cyclic (loop) graphs demonstrate superior graph-recovery performance (lower SHD, higher TPR, lower FDR) and greater physical consistency than state-of-the-art baselines, including in nonstationary settings. The approach offers a principled route to physics-informed causal reasoning in complex dynamical systems and suggests several avenues for extending causal discovery to richer mechanistic forms and spatiotemporal data.
Abstract
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.