Disentangling Continuous-Time Latent Dynamics: Identifiability of Latent SDEs via Diffusion Shifts

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.