Interpretable time series analysis with Gumbel dynamics

Learning identifiable representations in deep generative models remains a fundamental challenge, particularly for sequential data with regime-switching dynamics. Existing approaches establish identifiability under restrictive assumptions, such as stationarity or limited emission models, and typically rely on variational autoencoder (VAE) estimators, which introduce approximation gaps that limit the recovery of the latent structure. In this work, we address both the theoretical and practical limitations of this setting. First, we establish identifiability of a broad class of recurrent nonlinear switching dynamical systems under flexible assumptions, significantly extending prior results. Second, we introduce $Ω$SDS, a flow-based estimator that enables exact likelihood optimization using expectation-maximisation. Through empirical validation on both synthetic and real-world data, our results demonstrate that $Ω$SDS achieves improved disentanglement compared to VAE-based estimators and more accurate forecasting of underlying dynamics.

Learning system dynamics from observations is a critical problem in many applications over various real-world complex systems, e.g., climate, ecology, and fluid systems. Recently, neural dynamics modeling method have become a prevalent solution that embeds the object's observations into a latent space before learning dynamics using neural methods such as neural Ordinary Differential Equations (ODE). Existing dynamics modeling methods induce a specific model for each observation of different complex systems, resulting in poor generalization across systems. Inspired by the great success of pre-trained models, we conduct a generalized Pre-trained Dynamics EncoDER (PDEDER) which can embed the original state observations into a latent space where the dynamics can be captured more easily. To conduct the generalized PDEDER, we pre-train any Pre-trained Language Model (PLM) by minimizing the Lyapunov exponent objective, which constrains the chaotic behavior of governing dynamics learned in the latent space. By penalizing the divergence of embedded observations, our PDEDER promotes locally stable and well-structured latent dynamics, thereby facilitating more effective dynamics modeling than in the original observation space. In addition, we incorporate reconstruction and forecasting objectives to mitigate the risk of obtaining an over-smoothed latent space. Specifically, we collect 152 sets of real-world and synthetic observations from 23 complex systems as pre-training corpora and employ them to pre-train PDEDER. Given any future dynamic observation, we can fine-tune PDEDER with any specific dynamics modeling method. We evaluate PDEDER on 12 dynamic systems by short/long-term forecasting under both in-domain and cross-domain settings, and the empirical results indicate the effectiveness and generalizability of PDEDER.

In many domains, practitioners seek models that produce accurate forecasts while faithfully capturing latent system dynamics. Existing approaches typically sacrifice one of these goals: deep state space models often assume Gaussian latent transitions, limiting fit and forecasting, while diffusion models are highly expressive but lack principled inference for the underlying dynamics. To combine the strengths of both, we introduce the Diffusion-Driven State Space Model (DDSSM), which replaces the conventional Gaussian transition distribution with a diffusion model. Our DDSSM resolves the open problem of how to jointly train an autoencoder and a diffusion model on sequential data, thereby extending the literature on latent diffusion models for time series. Moreover, we find that the DDSSM empirically outperforms a state-of-the-art deep SSM at fitting and forecasting a simulated time series with multimodal transitions.