Search for: All records

Not all the information in a turbulent field is relevant for understanding particular regions or variables in the flow. Here, we present a method for decomposing a source field into its informative$$\boldsymbol {\varPhi }_{I}(\boldsymbol {x},t)$$and residual$$\boldsymbol {\varPhi }_{R}(\boldsymbol {x},t)$$components relative to another target field. The method is referred to as informative and non-informative decomposition (IND). All the necessary information for physical understanding, reduced-order modelling and control of the target variable is contained in$$\boldsymbol {\varPhi }_{I}(\boldsymbol {x},t)$$, whereas$$\boldsymbol {\varPhi }_{R}(\boldsymbol {x},t)$$offers no substantial utility in these contexts. The decomposition is formulated as an optimisation problem that seeks to maximise the time-lagged mutual information of the informative component with the target variable while minimising the mutual information with the residual component. The method is applied to extract the informative and residual components of the velocity field in a turbulent channel flow, using the wall shear stress as the target variable. We demonstrate the utility of IND in three scenarios: (i) physical insight into the effect of the velocity fluctuations on the wall shear stress; (ii) prediction of the wall shear stress using velocities far from the wall; and (iii) development of control strategies for drag reduction in a turbulent channel flow using opposition control. In case (i), IND reveals that the informative velocity related to wall shear stress consists of wall-attached high- and low-velocity streaks, collocated with regions of vertical motions and weak spanwise velocity. This informative structure is embedded within a larger-scale streak–roll structure of residual velocity, which bears no information about the wall shear stress. In case (ii), the best-performing model for predicting wall shear stress is a convolutional neural network that uses the informative component of the velocity as input, while the residual velocity component provides no predictive capabilities. Finally, in case (iii), we demonstrate that the informative component of the wall-normal velocity is closely linked to the observability of the target variable and holds the essential information needed to develop successful control strategies. 

Predicting extreme events in chaotic systems, characterized by rare but intensely fluctuating properties, is of great importance due to their impact on the performance and reliability of a wide range of systems. Some examples include weather forecasting, traffic management, power grid operations, and financial market analysis, to name a few. Methods of increasing sophistication have been developed to forecast events in these systems. However, the boundaries that define the maximum accuracy of forecasting tools are still largely unexplored from a theoretical standpoint. Here, we address the question: What is the minimum possible error in the prediction of extreme events in complex, chaotic systems? We derive the minimum probability of error in extreme event forecasting along with its information-theoretic lower and upper bounds. These bounds are universal for a given problem, in that they hold regardless of the modeling approach for extreme event prediction: from traditional linear regressions to sophisticated neural network models. The limits in predictability are obtained from the cost-sensitive Fano’s and Hellman’s inequalities using the Rényi entropy. The results are also connected to Takens’ embedding theorem using the information can’t hurt inequality. Finally, the probability of error for a forecasting model is decomposed into three sources: uncertainty in the initial conditions, hidden variables, and suboptimal modeling assumptions. The latter allows us to assess whether prediction models are operating near their maximum theoretical performance or if further improvements are possible. The bounds are applied to the prediction of extreme events in the Rössler system and the Kolmogorov flow.