Information-theoretic bounds in the prediction of extreme events and applications to turbulent flows

Predicting extreme events in turbulent flows, characterized by rare but intense fluctuations in flow properties, is of paramount importance due to their potential impact on the performance and reliability of a wide range of engineering systems. Various methods have been explored for predicting extreme events in turbulence; however, the theoretic bounds on the predictive accuracy of forecasting tools remain relatively unknown. Information theory, i.e., the science of message communication, offers a rigorous framework for investigating the fundamental limitations for the prediction and modeling of extreme events. Here, we leverage information-theoretic Fano-type inequalities to establish the bounds for predicting and modeling extreme events in turbulent flows. By investigating the inherent uncertainties and constraints that hinder predictive capabilities, Fano-type inequalities provide a lower bound on the probability of error over all the possible models. These bounds are universal and independent of the particular modeling tool employed. We demonstrate the application of the information-theoretic limits in the prediction of extreme events in a minimal turbulent channel flow. The time signals are defined as the energy contained in different Fourier modes of the three velocity components. The theoretical bounds on the errors are calculated for different variables and compared with actual models trained using numerical data. Our approach allows us to evaluate whether models are operating near their theoretical limits or whether further improvements are theoretically possible.