Data-Driven Identification of Stochastic Dynamical Systems

Abstract

The identification of stochastic dynamical systems from observational data represents one of the most significant challenges in modern applied mathematics and engineering. This comprehensive review examines the state-of-the-art in data-driven methods for discovering governing equations, estimating parameters, and predicting the behavior of complex systems subject to random perturbations. We present a systematic analysis of key methodologies including Sparse Identification of Nonlinear Dynamics (SINDy), Dynamic Mode Decomposition (DMD) and its extensions, Koopman operator theory, neural ordinary differential equations (Neural ODEs), and Bayesian inference approaches. Each method is evaluated in terms of its theoretical foundations, computational requirements, robustness to noise, and applicability to different classes of stochastic systems. Through extensive numerical experiments and real-world case studies, we demonstrate the relative strengths and limitations of each approach. Our findings reveal that while no single method dominates across all scenarios, hybrid approaches combining physics-informed constraints with machine learning offer the most promising path forward. We conclude with a discussion of open challenges and future research directions, including real-time identification, uncertainty quantification, and the integration of multi-fidelity data sources. This work provides researchers and practitioners with a comprehensive framework for selecting and implementing appropriate identification methods for stochastic dynamical systems.