An Online Gradient Flow Method for Optimizing the Statistical Steady-State Time Averages of Unsteady Turbulent Flows

View the PDF file from the paper entitled OGF: The method of online gradient to improve the average time of statistical statistical condition for unstable turbulent flows, by Tom Hickling and 3 other authors

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a summary:The turbulent flows are chaotic and unstable, but its statistical distribution turns into a statistical statistical condition. The engineering quantities of interest usually take the form of average time statistics such as $ \ Fra {1} {t} \ int_0^TF (u (x, \ tau; \ theta)) d \ tau \ overrustt {t \ rightarrow \ infty} {\ \ \ rightarrow} F (x; \ theta) $ ; \ Theet) $ are solutions for Navier- Stokes Equations with Parameters $ \ Theet $. When improving more than $ f (x; \ theet) $ has many engineering applications including engineering improvement, flow control, and closing modeling. However, this still represents an open challenge, as the current mathematical methods are unable to limit into actual representative numbers of network points. The main obstacle is the chaos in the turbulent flows: the calculated gradients in the neighboring way are diverged greatly like $ T \ rightarrow \ Infty $.

We develop a new way to flow online (OGF) that is largely developed into large systems of freedom and enables improvement for stable statistics of unstable chaotic simulation and disturbance solution. The online estimation method is determined to include $ f (x; \ theet) $ during the online updates simultaneously for parameters $ \ Theeta $. The main feature is the full online nature of the algorithm to facilitate the progress of improvement faster and total with the limited teams to avoid deviation of gradients due to chaos. The proposed OGF method for improvements is clarified on three chaotic and partial differential equations: Lorenz-63 equation, the Curamoto-Sephansky’s equation, and Navier-Correlation Solutions concludes in each case, the OGF method successfully works to reduce loss based on $ f (x; \ theta) $ by several orders in terms of size and recovery of teachers The best accurately.