May 2026· African Multidisciplinary Journal of Sciences and Artificial Intelligence· 0 citations· 64 references
Abstract
Identifying stochastic dynamical systems from observational data remains a major challenge in applied mathematics and engineering, particularly when complex systems are influenced by random perturbations and incomplete empirical information. This comprehensive review aims to examine state-of-the-art data-driven methods for discovering governing equations, estimating parameters, and predicting the behavior of stochastic dynamical systems. The review systematically analyzes key methodological approaches, including Sparse Identification of Nonlinear Dynamics (SINDy), Dynamic Mode Decomposition (DMD) and its extensions, Koopman operator theory, neural ordinary differential equations, and Bayesian inference. Each approach is evaluated in terms of its theoretical foundations, computational requirements, robustness to noise, and applicability to different classes of stochastic systems. Drawing on numerical experiments and real-world case studies, the findings show that no single method consistently outperforms others across all scenarios. Instead, hybrid approaches that integrate physics-informed constraints with machine learning demonstrate the strongest potential for advancing data-driven system identification. The review concludes that future research should address real-time identification, uncertainty quantification, and the integration of multi-fidelity data sources to improve the reliability and scalability of stochastic system modeling. This work contributes a comprehensive framework for guiding researchers and practitioners in selecting and implementing appropriate identification methods for stochastic dynamical systems.
The data-driven discovery of governing equations for dynamical systems has emerged as a transformative paradigm, enabling the extraction of interpretable and generalizable models from observational data. While modern techniques have advanced this field, traditional subset regression remains a foundational yet underutilized tool due to its reliance on uncorrelated residuals, a requirement often violated by time-series data. In this work, we revisit subset regression to identify dynamical systems governed by ordinary differential equations (ODEs), partial differential equations (PDEs), and differential algebraic equations (DAEs). We propose subset regression with known number of active features (sub-KNAFE), a user-determined sparsity mechanism that flexibly adapts to various complex nonlinear systems, while retaining the computational efficiency and inherent interpretability of traditional subset regression. We integrate sub-KNAFE with the SINDy framework, overcoming the limitation of subset regression in dynamical system identification. Numerical tests across a range of signal-to-noise ratios and dataset sizes demonstrate sub-KNAFE's superior noise robustness and data efficiency. Practical utility for sub-KNAFE is validated on two real-world datasets: the classic Lynx-Hare ecological population data and the ISO New England power system dataset, demonstrating its strong potential for practical deployment in scientific discovery and engineering applications.
Weizhen Li, Qiang Fu, Yifan Hong et al.· Scientific Reports· 0 citations
It is shown that the method consistently identifies compact governing equations while maintaining strong long-horizon predictive accuracy across canonical nonlinear systems and representative industrial processes, even under noisy and distribution-shifted data.
System identification provides the data-to-model link for analysis, prediction, diagnosis, and control of dynamical systems. As engineering and scientific systems become increasingly nonlinear, high-dimensional, networked, time-varying, and partially observed, identification methods must handle noisy data, incomplete prior knowledge, safety constraints, and downstream control requirements. This review surveys system identification for complex dynamical systems through a unified framework that connects model structure, operating mode, estimation target, and computational setting. We first revisit single-system identification, including regularized regression, sparse Bayesian learning, Kalman-type filtering, and recent structured recursive methods such as auxiliary-model, multi-innovation, hierarchical, filtering-based, and coupled identification. We then review multi-system identification for families of interacting or related subsystems, covering deep neural architectures, sparse equation discovery for ordinary and partial differential equations, hybrid and topology-aware modeling, and meta-learning-based fast adaptation. Finally, we discuss large language model-assisted workflows for symbolic regression, equation discovery, feature generation, and scientific modeling. By comparing these methodological streams, this review highlights their strengths, limitations, and applicability to prediction, synchronization, diagnosis, and control. We identify key open challenges, including robust identification under biased or limited data, uncertainty-aware modeling, interpretable learning, reproducible benchmarking, and tighter integration between identification, experiment design, and control synthesis.
Xiaoyu Zhang, Zi-Qiang Li, Ruizhe Shi et al.· Complex Engineering Systems· 0 citations
Data-driven approaches with formal guarantees have recently emerged as a powerful means for the verification and controller synthesis of complex dynamical systems. Interest in these methods is rapidly growing, as system models are often unavailable in practice, and challenges such as nonlinear behavior, uncertainty, and the curse of dimensionality typically render accurate modeling infeasible. These difficulties motivate leveraging limited data collected from the system while still providing formal guarantees on its overall behavior. The community has therefore proposed a few hundred articles on the development of data-driven frameworks that enable the formal verification and synthesis of dynamical systems without explicit models, addressing complex specifications beyond stability. Despite this rapid growth, existing results remain scattered and lack a coherent organization, limiting a clear understanding of their principles, distinctions, and practical potential. This survey fills this gap by providing a comprehensive overview of these data-driven methods for both deterministic and stochastic dynamical systems. We structure the literature around three main methodological pillars in formal methods: (in)finite-abstraction-based techniques, functional certificate approaches, such as control barrier certificates, and compositional methods. For each of these approaches, we classify the resulting data-driven guarantees into three main categories: (i) statistical guarantees grounded in probably approximately correct and scenario-based frameworks, (ii) guarantees derived from Lipschitz continuity, and (iii) guarantees exploiting structural properties. While the literature on deterministic systems is considerably richer, we also devote particular attention to the stochastic counterpart, highlighting the inherent differences and challenges that arise compared to the deterministic case.
Behrad Samari, Alessandro Abate, A. Girard et al.· 0 citations
Discovering dynamical systems from trajectory data is a central problem in applied mathematics and engineering. Whilst recent advances in machine learning have led to strong progress in data-driven system identification, much less attention has been given to systems with discontinuous dynamics. These systems are nevertheless highly relevant in applications, including climate dynamics and mechanical systems with friction. In this work, we consider the problem of identifying piecewise-smooth dynamical systems directly from trajectory data. Compared with the smooth setting, this requires recovering the governing equations and detecting the switching hyperplanes that separate different dynamical regimes and characterising their behaviour, such as sliding motion. We present a modular framework for discovering such systems by first estimating switching hyperplanes from data and then learning smooth dynamics within each region using geometry-constrained neural networks. The geometry-learning phase is studied from a statistical perspective, analysing the identifiability of the discontinuities and the robustness of the procedure. We also introduce a novel neural network architecture with a prescribed discontinuity set, and provide a theoretical analysis of its approximation properties. The approach is tested on low-dimensional benchmark problems, including dry-friction oscillators and the PP04 climate model for the ice ages.
D. Murari, Erik Jansson, Chris Budd Obe et al.· 0 citations
Abstract.
Population dynamics in fields such as molecular biology, epidemiology, and ecology exhibit highly stochastic and nonlinear behavior. In gene regulatory systems in particular, oscillations and multistability are especially common. Despite this, none of the currently available stochastic models for population dynamics are both accurate and computationally efficient for long-term predictions. A prominent model in this field, the linear noise approximation (LNA), is computationally efficient for tasks such as simulation, sensitivity analysis, and parameter estimation; however, it is only accurate for linear systems and short-time predictions. Other models may achieve greater accuracy across a broader range of systems, but they sacrifice computational efficiency and analytical tractability. This paper demonstrates that, with specific modifications, the LNA can accurately capture nonlinear dynamics in population processes. We introduce a new framework based on center manifold theory, a classical concept from nonlinear dynamical systems. This approach enables the identification of simple, system-specific modifications to the LNA, tailored to classes of qualitatively similar nonlinear dynamical systems. With these modifications, the LNA can achieve accurate long-term simulations without compromising computational efficiency. We apply our methodology to classes of oscillatory and bistable systems and present multiple examples from molecular population dynamics that demonstrate accurate long-term simulations alongside significant improvements in computational efficiency.
Frederick Truman-Williams, G. Minas· SIAM Journal on Applied Dyna...· 0 citations