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Neural networks for the prediction of chaos and turbulence



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Racca, Alberto 


Chaos is a deterministic, yet unpredictable, phenomenon that appears in multiple engineering applications. Predicting chaotic dynamics is challenging because infinitesimal perturbations to the system’s initial conditions grow exponentially in time. To further add complexity to the phenomenon, chaotic systems may come in different forms, such as extreme and spatiotemporal (turbulent) dynamics, which have different properties that require different approaches. When the governing equations of such systems are not known or computationally expensive to solve, data-driven methods are an effective tool to predict the evolution in time of the system. In this thesis, we develop data-driven methods for the prediction of prototypical, extreme and spatiotemporal chaos. The methods are based on echo state networks (ESNs), which are versatile recurrent neural networks that learn temporal correlations within data.

In the first part of the thesis, we focus on the network's hyperparameters, which markedly affect the performance of the machine. We optimise the procedure to select the hyperparameters, i.e. the validation strategy, in order to improve performance and robustness. First, we investigate common validation strategies, such as the single shot validation. Second, we propose the recycle validation, which is a computationally cheap validation strategy that is tailored for the prediction of chaotic and quasiperiodic dynamics with recurrent neural networks. Third, we compare Bayesian optimisation with the traditional grid search for optimal hyperparameter selection. The proposed validation strategies are shown to be more robust and outperform the state-of-the-art validation strategies in benchmark testcases.

In the second part of the thesis, we apply the optimised networks to predict and control extreme events in a chaotic flow. First, we employ binary classification metrics to analyse (i) how many of the extreme events predicted by the network actually occur in the test set (precision), and (ii) how many extreme events are correctly predicted by the network (recall). Second, we focus on the time-accurate prediction of extreme events. We show that echo state networks are able to predict extreme events up to more than five Lyapunov times before the events occur. Third, we analyse the statistical prediction of extreme events. By training the networks with datasets that contain non-converged statistics, we show that the networks are able to extrapolate the flow's long-term statistics. In other words, the networks are able to improve the statistical knowledge of the system with respect to the training data. Fourth, we design a simple and effective control strategy to prevent extreme events from occurring. The control strategy decreases the occurrence of extreme events up to one order of magnitude with respect to the uncontrolled system. Fifth, we analyse the performance of the networks for different Reynolds numbers, and show that the networks perform well across a wide range of regimes.

In the third part of the thesis, we tackle turbulence. We combine the optimised echo state networks with a convolutional autoencoder (CAE) into the convolutional autoencoder echo state network (CAE-ESN) to predict two-dimensional flows. The architecture computes the latent space, which is the manifold onto which the turbulent dynamics live, through a series of nonlinear filtering operations performed by the CAE; and then it predicts the time evolution of the turbulent state in the latent space via the ESN. We show that the CAE-ESN (i) finds a latent-space representation of the turbulent flow that has less than 1% of the degrees of freedom than the physical space; (ii) time-accurately and statistically predicts the flow in both quasiperiodic and turbulent regimes; and (iii) takes less than 1% of computational time to predict the turbulent flow than solving the governing equations.

This thesis enables the data-driven prediction of prototypical, extreme and spatiotemporal chaos, with a focus on fluid dynamics. Because of the ubiquity of chaos and generality of the methods, these techniques can be straightforwardly extended to other testcases of engineering interest.





Magri, Luca


Chaos, Convolutional neural networks, Echo state networks, Recurrent neural networks, Turbulence


Doctor of Philosophy (PhD)

Awarding Institution

University of Cambridge
Engineering and Physical Sciences Research Council (2275537)