Preamble to moment equations-driven identification for stochastic systems

This study presents a stochastic system identification framework based on moment equations for discovering the governing nonlinear stochastic differential equations (SDEs) directly from data. The framework is complemented by a Selective Segmented Sampling data-acquisition strategy that enables reliable estimation of the required statistical moments, even from limited experimental measurements. Reduced-order modeling of nonlinear dynamical systems from first principles remains a central challenge across science and engineering. Recent advances in data-driven discovery and machine learning have enabled direct identification of governing equations from data. However, most existing approaches are restricted to deterministic systems, providing no means to discover the underlying stochastic dynamics other than additive measurement noise. Inspired by sparse regression, the proposed method seeks to identify Langevin-type stochastic dynamics from observed data. The drift and diffusion functions are represented by a linear combination of candidate nonlinear functions with sparse matrices of coefficients. By deriving deterministic moment-evolution equations from the SDEs, the identification problem reduces to a sparse nonlinear regression, in which the sparse coefficients are identified by minimizing the discrepancy between predicted and observed time derivatives of the moments. The proposed framework is validated on several representative examples of increasing complexity, including stochastic Duffing oscillators, the van der Pol oscillator under additive noise, and the stochastic Mathieu equation. The results demonstrate excellent accuracy and robustness in recovering both deterministic and stochastic components of nonlinear dynamics, highlighting the method’s potential for data-driven discovery of stochastic systems in practical engineering applications.