How is the global energy challenge related to chaos and machine learning?

Energy production is arguably one of the most important factors underlying modern civilisation. Energy allows us to inhabit inhospitable parts of the Earth in relative comfort (using heating and air conditioning), create large cities (by efficiently transporting food and pumping water), or maintain our health (providing the energy for water purification). It also connects people by allowing long-distance travel and facilitating digital communication.

But energy is a sensitive subject at the moment, mainly for two reasons. Firstly, the way we currently produce energy is not sustainable: the Earth’s oil, coal, gas, and uranium reserves are finite, and we are tearing through them. Secondly, it is widely acknowledged that burning fossil fuels is affecting the Earth’s climate as we release greenhouse gases into the atmosphere. How we deal with these issues is a vital, but challenging problem.

Tokamaks are nuclear fusion reactors which are designed to prove the feasibility of fusion as a large-scale and carbon-free source of energy. These reactors are suggested as one of the potential solutions to the global energy challenge. Nuclear fusion involves controlling plasmas at temperatures of 100 Million degrees Celsius, which is ten times the temperature of the Sun. However, this produces unwanted turbulence in the tokamak due to the huge temperature gradients at certain plasma parameters. One of the challenges for Culham Centre for Fusion Energy (CCFE) is to identify such chaotic scenarios in order to avoid damage to the facility and to optimise the efficiency of energy production by stabilising the plasma.

Attracted by the recent spectacular successes of machine learning techniques for image classification, Debasmita Samaddar, a computational plasma physicist from CCFE, approached Oxford Mathematicians Nicolas Boullé, Vassilios Dallas, and Yuji Nakatsukasa to investigate whether machine learning can be employed to effectively control fusion reactors. The research that was carried out focused on how time series can be classified into chaotic or not (see Fig. 1) using machine learning.

Figure 1: A non-chaotic (left) and a chaotic (right) time series generated by the Lorenz system.

Contrary to standard machine learning techniques, the neural network was trained on a different and simpler set than the testing set of interest in order to demonstrate the generalisation ability of neural networks in this classification problem. The main challenge is to learn the chaotic features of the training set, without overfitting, and generalise on the testing data set, which behaves differently. Using a neural network that was trained on the Lorenz system, which is a system of three coupled nonlinear Ordinary Differential Equations (ODEs), we were able to classify time series of the Kuramoto-Sivashinsky (KS) equation  (see Fig. 2) as chaotic or not with high accuracy. The KS equation arises in a wide range of physical problems including instabilities in plasmas and is a characteristic example of a nonlinear PDE that exhibits spatiotemporal chaos.

 

Figure 2: A spatiotemporal chaotic solution of the Kuramoto-Sivashinsky equation (left) and its corresponding chaotic energy time series (right).

This important scientific result from this cross-disciplinary collaboration (facilitated by the Industrially Focused Mathematical Modelling Centre for Doctoral Training in Oxford) suggests that neural networks are able to identify the critical regimes that a fusion reactor might exhibit, paving the way to resolve central problems about the stability of CCFE's fusion reactors. It will be of great interest to see whether this work proves to be vital for the design of the next generation fusion reactors, helping them provide a sustainable energy solution.