This review surveys system identification for complex dynamical systems through a unified framework that connects model structure, operating mode, estimation target, and computational setting and identifies 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.
Abstract
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.
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
An interpretable identification framework based on universal neural ordinary differential equations (UNODEs), symbolic regression, and parameter refinement is developed that is competitive on autonomous polynomial systems and more effective in recovering compact symbolic structures for non-polynomial and explicitly time-varying dynamics.
Qing-Tong Dong· Engineering Research Express· 0 citations
State estimation is a fundamental component of model-based Fault Detection and Diagnosis (FDD) in dynamic systems, underpinning real-time monitoring, predictive maintenance, and safety-critical operations across industries such as aerospace, power systems, robotics, and autonomous vehicles. Traditional estimators, including the Kalman Filter (KF) and its variants, provide physically interpretable residuals for fault detection but often fail to deliver reliable performance under nonlinear dynamics, modeling uncertainties, sensor faults, and non-Gaussian noise. This paper presents a comprehensive review of state estimation-based FDD approaches, with a particular focus on Artificial Intelligence (AI)-augmented Kalman filtering and hybrid frameworks that integrate Machine Learning (ML) models, including Neural Networks (NNs), Support Vector Machines (SVMs), and Gaussian Processes (GPs), with classical estimation theory. The review systematically evaluates model-based, data-driven, and hybrid methods, comparing their robustness, accuracy, computational efficiency, scalability, and interpretability in complex Cyber-Physical Systems (CPSs). Furthermore, emerging trends and open research challenges are identified, including online adaptation, fault-tolerant estimation, sensor fusion, explainable artificial intelligence (XAI), and deployment in Industry 4.0 and Internet of Things (IoT)-enabled environments. By bridging classical estimation theory with modern AI techniques, this review provides a roadmap for designing intelligent, adaptive, and resilient FDD systems capable of enhancing reliability, operational safety, and real-world applicability.
Sahar Gargouri, Majdi Mansouri, Ahmed Anis Kahloul et al.· Energies· 0 citations
Data-driven modelling of nonlinear dynamical systems based on machine learning is reviewed in this paper. Nonlinear dynamics refer to all sorts of complicated and irregular phenomena in mathematics, physics, engineering and life that are not well explained by linear models. Based on the accumulation of observation data from sensors, experiments and high-fidelity simulations, machine learning has recently begun to be applied in system identification, state estimation and the discovery of governing equations. This paper systematically presents the basic ideas of dynamical systems and then introduces several data-driven methods, such as Sparse Identification of Nonlinear Dynamics (SINDy), Neural Ordinary Differential Equations (Neural ODEs) and Koopman operator theory. Based on the above research, a number of methods have been proposed to solve the problems of noise, irregular sampling, etc., in high-dimensional chaos, and their respective advantages are introduced below. At present, the problems of poor data quality, partial observability and lack of interpretability in deep learning models are well-known. In the future, Physics-Informed Machine Learning, uncertainty quantification and robust model methods will be applied in engineering. Data science and applied mathematics will be combined in this paper to offer a comprehensive introduction to the problems and models of complex systems.
Ming-Yang Wang· Theoretical and Natural Scie...· 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
To address the issue of decreased prediction reliability and control stability caused by model drift, non-stationary disturbances, and structural evolution in complex engineering systems, this paper proposes an adaptive digital twin-driven performance prediction and collaborative model update framework. This framework is based on a non-autonomous stochastic dynamic system, modeling the digital twin parameter field as a stochastic process in a time-varying function space. Simultaneously, as an optimization, stochastic differential equations and stochastic partial differential equations jointly characterize the coupled evolution mechanism of continuous disturbances, sudden shocks, and parameter drift. Based on this, a prediction-control coupled operator semigroup model is constructed. Nonlinear operator kernel expansion and spectral consistency constraints are introduced to maintain long-term consistency between the twin prediction operator and the dynamic structure of the physical system. Subsequently, a non-stationary Bayesian hierarchical prediction model and its variational recursive update mechanism are proposed. Furthermore, the digital twin model and control model are abstracted into a two-agent collaborative game system. This model constructs a continuous-time policy gradient flow and cross-regularized co-evolution equation. Through a demand-driven state transition operator and a task-level collaborative control mechanism, the framework's adaptability in multi-task and multi-scenario environments is further enhanced. Systematic results show that, compared with the current best comparative method, the proposed method reduces the mean squared error (MSE) and mean absolute error (MAE) by approximately 23.5% and $14.9{{\% }}$, respectively, in long-term prediction tasks.
Hongjuan Zhu· 2026 International Conferenc...· 0 citations
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