Grad-Shafranov equilibria via data-free physics informed neural networks
Byoungchan Jang Alan A. Kaptanoglu Rahul Gaur Shaowu Pan Matt Landreman William Dorland
A large number of magnetohydrodynamic (MHD) equilibrium calculations are
often required for uncertainty quantification, optimization, and real-time
diagnostic information, making MHD equilibrium codes vital to the field of
plasma physics. In this paper, we explore a method for solving the
Grad-Shafranov equation by using Physics-Informed Neural Networks (PINNs). For
PINNs, we optimize neural networks by directly minimizing the residual of the
PDE as a loss function. We show that PINNs can accurately and effectively solve
the Grad-Shafranov equation with several different boundary conditions. We also
explore the parameter space by varying the size of the model, the learning
rate, and boundary conditions to map various trade-offs such as between
reconstruction error and computational speed. Additionally, we introduce a
parameterized PINN framework, expanding the input space to include variables
such as pressure, aspect ratio, elongation, and triangularity in order to
handle a broader range of plasma scenarios within a single network.
Parametrized PINNs could be used in future work to solve inverse problems such
as shape optimization.