In magnetic confinement fusion research, the achievement of high plasma pressure is key to reaching the goal of net energy production. The magnetohydrodynamic (MHD) model is used to self-consistently calculate the effects the plasma pressure induces on the magnetic field used to confine the plasma. Such MHD calculations-usually done computationally-serve as input for the assessment of a number of important physics questions. The variational moments equilibrium code (VMEC) is the most widely used to evaluate 3D ideal-MHD equilibria, as prominently present in stellarators. However, considering the computational cost, it is rarely used in large-scale or online applications (e.g. Bayesian scientific modeling, real-time plasma control). Access to fast MHD equilibria is a challenging problem in fusion research, one which machine learning could effectively address. In this paper, we present artificial neural network (NN) models able to quickly compute the equilibrium magnetic field of Wendelstein 7-X. Magnetic configurations that extensively cover the device operational space, and plasma profiles with volume-averaged normalized plasma pressure β (β = 2μ 0 p B 2 ) up to 5% and non-zero net toroidal current are included in the data set. By using convolutional layers, the spectral representation of the magnetic flux surfaces can be efficiently computed with a single network. To discover better models, a Bayesian hyper-parameter search is carried out, and 3D convolutional NNs are found to outperform feed-forward fully-connected NNs. The achieved normalized root-mean-squared error, the ratio between the regression error and the spread of the data, ranges from 1% to 20% across the different scenarios. The model inference time for a single equilibrium is on the order of milliseconds. Finally, this work shows the feasibility of a fast NN drop-in surrogate model for VMEC, and it opens up new operational scenarios where target applications could make use of magnetic equilibria at unprecedented scales.

The computational cost of constructing 3D magnetohydrodynamic (MHD) equilibria is one of the limiting factors in stellarator research and design. Although data-driven approaches have been proposed to provide fast 3D MHD equilibria, the accuracy with which equilibrium properties are reconstructed is unknown. In this work, we describe an artificial neural network (NN) that quickly approximates the ideal-MHD solution operator in Wendelstein 7-X (W7-X) configurations. This model fulfils equilibrium symmetries by construction. The MHD force residual regularizes the solution of the NN to satisfy the ideal-MHD equations. The model predicts the equilibrium solution with high accuracy, and it faithfully reconstructs global equilibrium quantities and proxy functions used in stellarator optimization. We also optimize W7-X magnetic configurations, where desiderable configurations can be found in terms of fast particle confinement. This work demonstrates with which accuracy NN models can approximate the 3D ideal-MHD solution operator and reconstruct equilibrium properties of interest, and it suggests how they might be used to optimize stellarator magnetic configurations.

This article reviews applications of Bayesian inference and machine learning in nuclear fusion research. Current and next-generation nuclear fusion experiments require analysis and modelling efforts that integrate different models consistently and exploit information found across heterogeneous data sources in an efficient manner. Model-based Bayesian inference provides a framework well suited for the interpretation of observed data given physics and probabilistic assumptions, also for very complex systems, thanks to its rigorous and straightforward treatment of uncertainties and modelling hypothesis. On the other hand, machine learning, in particular neural networks and deep learning models, are based on black-box statistical models and allow the handling of large volumes of data and computation very efficiently. For this reason, approaches which make use of machine learning and Bayesian inference separately and also in conjunction are of particular interest for today’s experiments and are the main topic of this review. This article also presents an approach where physics-based Bayesian inference and black-box machine learning play along, mitigating each other’s drawbacks: the former is made more efficient, the latter more interpretable.