Bayesian neural network for plasma equilibria in the Korea Superconducting Tokamak Advanced Research

Abstract

Fusion-graded plasmas are one of the physically complex systems, resulting in continuous establishment of plasma theories for unclarified physical phenomena in order to thoroughly control nuclear fusion reactors. Deep learning has drawn vast attention to this field of controlled fusion plasma to link physical phenomena with control-relevant parameters without a deepened understanding about plasma theories. Albeit, quantifying the uncertainty of deep learning models has been constantly requested due to their fundamental shortage of physical understanding. Thus, a concept of a reliable deep learning model to be able to present their probability distributions is raised as well as a method to inculcate physical theories in the model is also concerned. These are the main concept focused in this thesis with a tokamak experiment, one of the nuclear fusion experiments by confining the plasmas via toroidal and poloidal magnetic fields in a torus shape device. Since the tokamak confines a plasma magnetically, balancing the Lorentz force due to the magnetic field with the plasma pressure is crucial. This balanced state with equilibrium assumption is called plasma equilibrium, giving us the shape and location of the plasma determined and controlled by the external coil currents of the tokamak. However, this plasma shape cannot be directly measured due to the harsh environment caused by the plasma itself of 100 million degrees Celsius, thus the shape is indirectly reconstructed from the force balance and Maxwell's equations consistent with externally and locally measured plasma information. The Grad-Shafranov (GS) equation derived from those equations is used to reconstruct the plasma. This equation is a two-dimensional second-order differential equation, inherently requiring numerical analysis so that human decisions such as selecting some of the measured signals arbitrarily for numerical convergence are followed. Furthermore, it is likely to sacrifice accuracy of solutions of the equation for real-time tokamak controls due to multiple iterations in numerical analysis which requires intensive computations. Although there were neural network based real-time approaches via supervised learning with databases from numerical algorithms, they were inevitably under the influence of human decisions. Hence, this thesis suggests a reconstruction method based on deep neural networks which are able to not only estimate their uncertainties but also learn the governing equation themselves without depending on previous numerical algorithms. Namely, our neural networks solve the GS equation via a unsupervised learning algorithm and show probability distributions of their solutions based on Bayesian neural networks. Since solving the GS equation is a free-boundary problem, our networks are supported by an auxiliary module that detects the plasma boundary from the network outputs. Furthermore, we introduce preprocessing methods for the network inputs to address the magnetic signal drift, the flux loop inconsistency and the magnetic signal impairment based on Bayesian inference, Gaussian processes and neural networks. These methods are developed to guarantee the use of the networks in any circumstance of the tokamak experiments. In addition, we also prove that the Grad-Shafranov equation can be used as a cost function of the networks with a given equilibrium database. The principles and methods applied here are not only acceptable for fusion research but also applicable to various engineering and scientific fields. Thus, we expect that our proposal which fulfills physical reliability for the use of deep learning deserves further studies for various complex physics systems.