state space modeling

This work addresses the challenge of modeling and forecasting stochastic nonlinear dynamical systems under noisy and partially observable conditions by proposing a deep spectral learning framework. The method employs a learnable neural encoder to construct Markovian latent states in a feature space, whose dynamics and observations are governed by learned transition and observation operators. It uniquely unifies spectral learning, Bayesian filtering, and Koopman mode decomposition within an end-to-end trainable architecture. Efficient state estimation is achieved through functional canonical correlation analysis, Galerkin projection, and a closed-form ridge-regularized solution. Experimental results demonstrate that the proposed approach significantly outperforms baseline methods—including sequential Bayesian filters and dynamic mode decomposition—across diverse scenarios, exhibiting strong robustness to both observation noise and partial observability.

This work addresses the challenge of learning compact linear state-space representations of nonlinear dynamical systems from observational data, circumventing the need to solve non-convex system identification problems directly. The authors propose a two-stage, provably correct pipeline: first, an implicit spectral predictor is learned via Observation Spectral Filtering (OSF), a convex optimization method; second, this predictor is distilled into an explicit linear dynamical system (LDS) through a spectral-to-LDS distillation procedure. This approach yields the first end-to-end provable linearization of nonlinear systems, with error bounds that depend only on observation complexity rather than latent dimensionality and exhibit exponentially small distillation error. Empirical results demonstrate that the resulting compact LDS predictors match or outperform baseline models trained directly on standard LDS benchmarks and MuJoCo behavioral cloning tasks.

To address the cyclic dependency problem arising from the coupling of latent states and nonlinear dynamics in time-series modeling, this paper proposes LaNoLem: a method that models the system as a time-varying dynamical process in a latent space and decouples latent-state inference from dynamics learning via an alternating minimization algorithm. It introduces a fully automated, human-in-the-loop-free complexity regularization criterion to enable adaptive control of model capacity. By jointly optimizing latent-state representation, nonlinear differential equation learning, and dynamics estimation, LaNoLem achieves state-of-the-art accuracy in dynamical system identification. Moreover, it significantly outperforms existing methods on multi-step long-horizon forecasting tasks—particularly for systems exhibiting intricate hidden mechanisms and long-range temporal dependencies.

Identifying latent states and decoupling dynamic modes in stochastic nonlinear dynamical systems remains challenging due to inherent nonlinearity and randomness. Method: This paper proposes a hidden Markov operator representation framework embedded in a reproducing kernel Hilbert space (RKHS). It uniquely integrates stochastic realization theory with kernel embedding techniques to jointly learn latent-space structure and state-transition operators; extends the Kalman filter to nonlinear stochastic systems; and introduces operator-theoretic dynamic mode decomposition (O-DMD). The technical pipeline encompasses spectral learning, neural-network-based adaptive kernel construction, generalized sequential state estimation, and operator spectral analysis. Results: Evaluated on synthetic and real-world datasets, the method significantly improves latent-state estimation accuracy, enables interpretable and decoupled extraction of dynamical modes, and establishes a unified, robust, and interpretable paradigm for modeling and inference in stochastic nonlinear systems.

This work addresses the challenge that traditional time-invariant models struggle to effectively capture switching dynamics in time-varying systems. To overcome this limitation, the authors propose a neural network–based time-varying state-space model that incorporates a learnable dictionary of time-varying basis functions. This design flexibly represents diverse temporal evolution patterns of system dynamics while maintaining manageable computational complexity, substantially enhancing the model’s capacity to capture switching sequences. The study further reveals an optimal allocation strategy for time-varying degrees of freedom across model components. Experimental results demonstrate that the proposed model consistently outperforms existing time-invariant approaches on both synthetic switching systems and speech denoising tasks, confirming its effectiveness and strong generalization capability.

State space models often pose significant challenges for parameter inference due to intractable likelihood functions. This work proposes Truncated Neural Likelihood Estimation (T-SNL), a novel approach that introduces a sequence truncation strategy to dramatically enhance training stability and computational efficiency while preserving estimation accuracy. By integrating simulation-based inference with Bayesian parameter learning, T-SNL enables amortized inference for long-sequence modeling and observation updates. Empirical evaluations demonstrate that T-SNL consistently outperforms existing methods in terms of sample efficiency, robustness, and scalability to long sequences, offering a flexible and efficient framework for parameter inference in complex dynamical systems.

This work addresses the challenge of rapid dynamic shifts in streaming time series caused by abrupt environmental changes or varying input delays. The authors propose a system tensor representation based on Markov parameter sequences, modeling the streaming data as a dynamic mixture of delay systems. By constructing fixed-length tensor summaries that jointly encode system dynamics and input–output delay characteristics, the method enables efficient compression and retrieval of historical patterns through tensor decomposition. Within an online learning framework, the system dynamically selects the optimal submodel to match the current state, achieving strong adaptability to nonstationary time series while maintaining low memory overhead. Experimental results on real-world datasets demonstrate that the proposed approach significantly outperforms existing methods in both prediction accuracy and adaptation speed, particularly under highly nonstationary conditions.

This study addresses the challenge of real-time modeling of complex nonlinear dynamical systems in nonstationary data streams. Building upon Koopman operator theory, the proposed approach embeds nonlinear dynamics into a reproducing kernel Hilbert space (RKHS) to obtain a linear representation and introduces a dual-view probabilistic latent variable model that jointly captures both raw observations and RKHS features. A statistical hypothesis testing mechanism is incorporated to enable adaptive detection of abrupt distributional shifts, triggering incremental parameter updates to maintain predictive accuracy while ensuring computational efficiency. Extensive experiments across 71 benchmark datasets from diverse domains demonstrate that the method significantly outperforms existing approaches in both real-time prediction accuracy and computational efficiency.

Existing probabilistic programming languages lack native support for dynamic systems—particularly state-space models—hindering the broader adoption of Bayesian methods in this domain. This work introduces dynestyx, a library that provides first-class, unified, and user-friendly support for state-space models within a probabilistic programming framework. dynestyx enables flexible specification of priors, accommodates both discrete- and continuous-time dynamics, handles mixed-effects data, and facilitates joint Bayesian inference over latent states and model parameters with full uncertainty quantification. By doing so, this contribution substantially enhances the accessibility, flexibility, and practical utility of dynamic system modeling across statistics, signal processing, and machine learning.