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.

翻译：大量磁流体动力学（MHD）平衡计算常用于不确定性量化、优化及实时诊断信息获取，这使得MHD平衡代码成为等离子体物理领域的关键工具。本文探索了一种利用物理信息神经网络（PINNs）求解Grad-Shafranov方程的方法。针对PINNs，我们通过直接最小化偏微分方程残差作为损失函数来优化神经网络。研究表明，PINNs能够在多种边界条件下准确高效地求解Grad-Shafranov方程。我们还通过改变模型规模、学习率及边界条件来探索参数空间，以映射重建误差与计算速度之间的权衡关系。此外，我们提出参数化PINN框架，将输入空间扩展至包含压强、纵横比、拉长率和三角变形度等变量，从而在单一网络中处理更广泛的等离子体场景。参数化PINNs未来可应用于形状优化等逆问题的求解。