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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]

Data-driven equation identification for dynamical systems has achieved great progress, which for static systems, however, has not kept pace. Unlike dynamical systems, static systems are time invariant, so we cannot capture discrete data along the time stream, which requires identifying governing equations only from scarce data. This work is devoted to this topic, building a data-driven method for extracting the differential-variational equations that govern static behaviors only from scarce, noisy data of responses, loads, as well as the values of system attributes if available. Compared to the differential framework typically adopted in equation identification, the differential-variational framework, due to its spatial integration and variation arbitrariness, brings some advantages, such as high robustness to data noise and low requirements on data amounts. The application, efficacy, and all the aforementioned advantages of this method are demonstrated by four numerical examples, including three continuous systems and one discrete system. [ABSTRACT FROM AUTHOR]