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Stochastic averaging, as an effective technique for dimension reduction, is of great significance in stochastic dynamics and control. However, its practical applications in industrial and engineering fields are severely hindered by its dependence on governing equations and the complexity of mathematical operations. Herein, a data-driven method, named data-driven stochastic averaging, is developed to automatically discover the low-dimensional stochastic differential equations using only the random state data captured from the original high-dimensional dynamical systems. This method includes two successive steps, that is, extracting all slowly varying processes hidden in fast-varying state data and identifying drift and diffusion coefficients by their mathematical definitions. It automates dimension reduction and is especially suitable for cases with unavailable governing equations and excitation data. Its application, efficacy, and comparison with theory-based stochastic averaging are illustrated through several examples, numerical or experimental, with pure Gaussian white noise excitation or combined excitations. [ABSTRACT FROM AUTHOR]

Canonical equations play a pivotal role in various sub-fields of physics and mathematics. However, for complex systems and systems without first principles, deriving canonical equations analytically is quite laborious or might even be impossible. This work is devoted to automatedly distilling the canonical equations solely from random state data. The random state data are collected from stochastically excited, dissipative dynamical systems either experimentally or numerically, while other information, such as the system characterization itself and the excitations, is not needed. The identification procedure comes down to a nested optimization problem, and the explicit expressions of the momentum (density) functions and energy (density) functions are identified simultaneously. Three representative examples are investigated to illustrate its high accuracy of identification, the small requirement for data amount, and high robustness to excitations and dissipation. The identification procedure serves as a filter, filtering out nonconservative information while retaining conservative information, which is especially suitable for systems with unobtainable excitations. [ABSTRACT FROM AUTHOR]