Grad-Shafranov equilibria via data-free physics informed neural networks (PINNs)

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). PINNs optimize neural networks by directly minimizing the residual of the PDE as a loss function. We use PINNs to show that they can accurately and effectively solve the Grad-Shafranov equation with Solov'ev profiles under several different boundary conditions. We also explore the parameter space by varying the size of the model, number of collocation points, and boundary conditions in order 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 like pressure P, inverse aspect ratio $varepsilon$, elongation $kappa$, and triangularity $delta$ in order to handle a broader range of plasma scenarios.