Surrogate models for plasma displacement and current in 3D perturbed magnetohydrodynamic equilibria in tokamaks

Abstract

A numerical database of over one thousand perturbed three-dimensional (3-D) equilibria has been generated, constructed based on the MARS-F (Liu Y.Q. et al 2000 Phys. Plasmas 7, 3681) computed plasma response to the externally applied 3-D field sources in multiple tokamak devices. Perturbed 3-D equilibria with the n=1-4 (n is the toroidal mode number) toroidal periodicity are computed. Surrogate models are created for the computed perturbed 3-D equilibrium utilizing model order reduction (MOR) techniques. In particular, retaining the first few eigenstates from the singular value decomposition (SVD) of the data is found to produce reasonably accurate MOR-representations for the key perturbed quantities, such as the perturbed parallel plasma current density and the plasma radial displacement. SVD also helps to reveal the core versus edge plasma response to the applied 3-D field. For the database covering the conventional aspect ratio devices, about 95% of data can be represented by the truncated SVD-series with inclusion of only the first 5 eigenstates, achieving a relative error below 20%. The MOR-data is further utilized to train neural networks (NNs) to enable fast reconstruction of perturbed 3-D equilibria, based on the 2-D equilibrium input and the 3-D source field. The best NN-training is achieved for the MOR-data obtained with a global SVD approach, where the full set of samples used for the NN training and testing are stretched and form a large matrix which is then subject to SVD. The fully connected multi-layer perceptron, with one or two hidden layers, can be trained to predict the MOR-data with less than 10% relative error. As a key insight, a better strategy is to train separate NNs for the plasma response fields with different toroidal mode numbers. It is also better to apply MOR and to subsequently train NNs separately for the conventional and low aspect ratio devices, due to enhanced toroidal coupling of Fourier spectra in the plasma response in the latter case.

Application of machine learning and artificial intelligence to extend EFIT equilibrium reconstruction

“…Similar level of accuracy is achieved for another key perturbed quantity associated with the perturbed 3D equilibrium, i.e. the radial displacement of the plasma [61].…”

“…The large amount of the 3D equilibrium data suggests that MOR techniques may play an important role. This serves two purposes: (i) to reduce the huge size of the data in the full 3D representation (by more than one order of magnitude), and more importantly (ii) to facilitate training of the ML algorithms for achieving real-time reconstruction of 3D equilibria [61], based on the 2D equilibrium input.…”

Validation of MARS-F modeling of plasma response to RMPs using internal measurements on DIII-D

“…The new post-processing routines established in this work will also help this kind of 3D equilibrium database study, which was recently initiated. 45 This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express, or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights.…”

Deep learning approaches to recover the plasma current density profile from the safety factor based on Grad–Shafranov solutions across multiple tokamaks

“…We achieve this by utilizing machine learning (ML) techniques based on a numerically generated equilibrium database. Machine learning has been widely used in fusion research, for example in plasma instability and disruption predictions [3][4][5][6][7][8][9][10][11][12][13][14], magnetic control [15], non-powerlaw scaling [16], turbulent transport [17][18][19] and plasma profile predictions [20,21], as well as fast magnetic equilibrium reconstruction [22][23][24][25][26][27].…”