Parametric system identification methods estimate the parameters of explicitly defined physical systems from data. Yet, they remain constrained by the need to provide an explicit function space, typically through a predefined library of candidate functions chosen via available domain knowledge. In contrast, deep learning can demonstrably model systems of broad complexity with high fidelity, but black-box function approximation typically fails to yield explicit descriptive or disentangled representations revealing the structure of a system. We develop a novel identifiability theorem, leveraging causal representation learning, to uncover disentangled representations of system parameters without structural assumptions. We derive a graphical criterion specifying when system parameters can be uniquely disentangled from raw trajectory data, up to permutation and diffeomorphism. Crucially, our analysis demonstrates that global causal structures provide a lower bound on the disentanglement guarantees achievable when considering local state-dependent causal structures. We instantiate system parameter identification as a variational inference problem, leveraging a sparsity-regularised transformer to uncover state-dependent causal structures. We empirically validate our approach across four synthetic domains, demonstrating its ability to recover highly disentangled representations that baselines fail to recover. Corroborating our theoretical analysis, our results confirm that enforcing local causal structure is often necessary for full identifiability.
Figures & tables
Figure 1: High-level overview of the developed theory. An observed trajectory (left) is encoded into a vector of latent system parameters (marked in dark blue). The developed theory shows that enforcing sparse causal relations between parameters and system components in the decoder (which performs one-step prediction) provably disentangles the system parameter representation.
Figure 2: Depiction of the ground-truth DAGs of the evaluation environments. The left and centre-left correspond to the dual particle and springs environments, respectively, and both satisfy the global graphical criterion for disentanglement. The centre-right and right graphs correspond to the local particle and bounce environments, respectively. Coloured arrows indicate causal edges that are only active in subsets of the state space, and only a limited number of subsets are shown for the bounce environment. The local particle and bounce environments satisfy only the local, not the global, graphical criterion for disentanglement.
Figure 3: Comparison of disentanglement across the test environments, where an MCC of 1.0 represents perfect disentanglement. All trials are repeated over eight random seeds. Box plots display the minimum, lower quartile, upper quartile, and maximum values. The validation reconstruction loss is shown at the bottom, indicating that all models are approximately equiperformant in these environments. The VCD baseline, which learns static graphs, strongly disentangles in the first two environments, which satisfies the global graph criterion. In contrast, only SPARTAN, which learns state-dependent graphs, consistently disentangles in all environments.
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, coordinate-wise scaling, and a possible constant shift, 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.
Yuanyuan Wang, Wenjie Wang, Haoxuan Li +2
Mohamed bin Zayed University of Artificial Intelligence · The University of Melbourne · Peking University +1
We study the problem of recovering the governing ODE of a dynamical system from unstructured, high-dimensional observations such as images. Existing methods for ODE discovery typically assume direct measurements of the variables, or do not provide theoretical guarantees on the learned variables and equations. While Causal Representation Learning (CRL) methods provide guarantees on identifying variables from high-dimensional observations up to component-wise diffeomorphisms, we show that in general these variables cannot be used directly as input to equation discovery methods, which typically assume that the variables will lead to sparse equations. So we introduce SParse Equivalent Equation Discovery AutoEncoder (SPEED-AE), a framework that combines a pretrained CRL method with a component-wise autoencoder that learns transformations of variables that are amenable to sparse ODE discovery. We show that for polynomial ODEs, this additional step allows us to restrict the identifiability of each variable from polynomial to monomial diffeomorphisms. Experiments on Lotka-Volterra, Lorenz, and a two-pendulum system show that SPEED-AE improves on the disentanglement of the CRL methods and that it recovers ODEs that are closest to the ground truth, while achieving state-of-the-art forecasting performance.
We provide explicit, finite-sample guarantees for learning causal representations from data with a sublinear number of environments. Causal representation learning seeks to provide a rigourous foundation for the general representation learning problem by bridging causal models with latent factor models in order to learn interpretable representations with causal semantics. Despite a blossoming theory of identifiability in causal representation learning, estimation and finite-sample bounds are less well understood. We show that causal representations can be learned with only a logarithmic number of unknown, multi-node interventions, and that the intervention targets need not be carefully designed in advance. Through a careful perturbation analysis, we provide a new analysis of this problem that guarantees consistent recovery of (a) the latent causal graph, (b) the mixing matrix and representations, and (c) unknown intervention targets.