Neural network tokamak equilibria with incompressible flows

TL;DR

This paper demonstrates the use of deep neural networks to solve the generalized Grad-Shafranov equation for axisymmetric plasma equilibria with flows, achieving accurate solutions and showcasing the interpolation abilities of ANNs in tokamak modeling.

Contribution

It introduces a neural network approach to solve the GGSE for plasma equilibria, including cases with flows and high confinement modes, with benchmarking against analytic solutions.

Findings

ANN solutions accurately match known analytic solutions.

Changing training point distribution minimally affects accuracy.

ANNs effectively interpolate entire magnetic configurations.

Abstract

We present several numerical solutions to a generalized Grad-Shafranov equation (GGSE), which governs axisymmetric plasma equilibria with incompressible flows of arbitrary direction, using fully connected, feed-forward, deep neural networks, also known as multi-layer perceptrons. Such artificial neural network (ANNs) are trained to approximate tokamak-relevant equilibria upon minimizing the GGSE mean squared residual in the plasma volume and the poloidal flux function on the plasma boundary. Solutions for the Solovev and the general linearizing ansatz for the free functions involved in the GGSE are obtained and benchmarked against known analytic solutions. We also construct a non-linear equilibrium incorporating characteristics relevant to the high confinement mode. In our numerical experiments it was observed that changing the radial distribution of the training points has a surprisingly small effect on the accuracy of the trained solution. In particular it is shown that localizing the training points at the plasma edge results in ANN solutions that describe quite accurately the entire magnetic configuration, thus demonstrating the interpolation capabilities of the ANNs.