Disentangling Continuous-Time Latent Dynamics: Identifiability of Latent SDEs via Diffusion Shifts
2026-06-26 • Machine Learning
Machine Learning
AI summaryⓘ
The authors address the challenge of understanding hidden causes in continuous-time systems modeled by stochastic differential equations (SDEs). They show that by using differences in noise levels across environments, it's possible to identify the hidden variables up to some simple transformations, even without strict assumptions about the system's behavior. Their results first apply to a specific type of linear system and then extend to more general cases, allowing recovery of the system's causal structure. They validate their theory with simulations and demonstrate it on real sensor data from a bridge.
causal representation learningstochastic differential equationsidentifiabilitylatent variablesdiffusion covarianceOrnstein–Uhlenbeck processnonlinear diffeomorphismcausal graphenvironment shiftslatent disentanglement
Authors
Yuanyuan Wang, Wenjie Wang, Haoxuan Li, Mingming Gong, Kun Zhang
Abstract
Causal representation learning for time series has developed strong identifiability results in discrete-time latent causal models, but identifiability in continuous-time latent stochastic differential equation (SDE) models remains largely open. We address this gap using environment-induced shifts in diffusion covariance. We study additive-noise latent SDEs observed through an unknown nonlinear diffeomorphism, with shared drift but environment-specific diffusion covariance. We show that two diagonal diffusion regimes with pairwise distinct coordinate-wise variance ratios identify the latent coordinates up to permutation and scaling, without any sparsity assumption on the drift. We first prove this result for linear Ornstein--Uhlenbeck systems and then extend it to general additive-noise latent SDEs. Under mild smoothness, the instantaneous drift-Jacobian causal graph is identifiable up to the same permutation. We propose a two-stage estimator for latent disentanglement and optional graph recovery; experiments on synthetic systems confirm the predicted identifiability boundary, and an application to Hardanger Bridge monitoring data illustrates the approach on real sensor trajectories.