A comprehensive overview of data-driven methods for both deterministic and stochastic dynamical systems, highlighting the inherent differences and challenges that arise compared to the deterministic case.
Abstract
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.
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.
Rishav Jha, Kameshwar Sahani, S. K. Sahani et al.· African Multidisciplinary Jo...· 0 citations
Analytical tools derived from nonlinear dynamics and dynamical systems theory, such as phase-space reconstruction and Recurrence Quantification Analysis (RQA), provide a powerful framework for investigating complex systems across different scientific domains. These methods allow the identification of dynamical structures, including recurrence, nonlinearity, and transitions between states, in time series data originating from diverse contexts. Scientific research is often shaped by two opposing forces that resemble the dynamics of physics: a centrifugal force, associated with increasing specialization, and a centripetal force, associated with interdisciplinarity. The rapid development of technologies and analytical methods has led to highly specialized languages and frameworks, which, while enabling scientific progress, can also generate fragmentation and communication barriers between disciplines. In contrast, interdisciplinarity emerges as a centripetal force that promotes the identification of shared analytical frameworks across domains. In this context, the transfer of methods is not merely a consequence of mathematical convenience but reflects the presence of common dynamical properties governed by similar physical principles. Artificial intelligence, integrated within physics-informed computational frameworks, provides a powerful tool for analyzing complex, high-dimensional, and heterogeneous datasets while preserving the dynamical structure of the underlying system. This convergence is not merely technical: the same nonlinear dynamical principles that govern physiological and cognitive systems appear to operate within artificial ones, suggesting that AI is not external to the phenomena this manuscript addresses but continuous with them. This inherent interdisciplinarity positions AI as a centripetal force, drawing together methods, languages, and findings from otherwise distant disciplines around a shared dynamical core.
Giovanna Zimatore, Piercesare Grimaldi, S. Hatzopoulos et al.· Frontiers of Physics· 0 citations
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
A physics-informed Koopman representation based on generalized momenta is introduced, yielding a linear control-affine model in lifted coordinates with known input structure that avoids the bilinear state – input coupling inherent in standard Koopman approaches, enabling improved prediction accuracy and tractable controller synthesis.
This work presents a modular framework for discovering piecewise-smooth dynamical systems by first estimating switching hyperplanes from data and then learning smooth dynamics within each region using geometry-constrained neural networks.
D. Murari, Erik Jansson, Chris Budd Obe et al.· 0 citations