# Grad-Shafranov equilibria via data-free physics informed neural networks

arXiv: https://arxiv.org/abs/2311.13491 · HTML: https://mlsanity.com/paper/2311.13491/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 38 references that are in the map.
- First posted on arXiv 22 Nov 2023.

## Builds on (most cited first, 12 of 12)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks](https://mlsanity.com/paper/2207.10289/) (2022)
- [Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10566/) (2017)
- [DAS-PINNs: A deep adaptive sampling method for solving high-dimensional partial differential equations](https://mlsanity.com/paper/2112.14038/) (2021)
- [The Old and the New: Can Physics-Informed Deep-Learning Replace Traditional Linear Solvers?](https://mlsanity.com/paper/2103.09655/) (2021)
- [Deep neural operators can serve as accurate surrogates for shape optimization: A case study for airfoils](https://mlsanity.com/paper/2302.00807/) (2023)
- [Demystifying Learning Rate Policies for High Accuracy Training of Deep Neural Networks](https://mlsanity.com/paper/1908.06477/) (2019)
- [Deep neural network Grad-Shafranov solver constrained with measured magnetic signals](https://mlsanity.com/paper/1911.02882/) (2019)
- [Neural net modeling of equilibria in NSTX-U](https://mlsanity.com/paper/2202.13915/) (2022)
- [Separable PINN: Mitigating the Curse of Dimensionality in Physics-Informed Neural Networks](https://mlsanity.com/paper/2211.08761/) (2022)

## Built on by (most cited first, 5 of 5)

- [Calibrated Physics-Informed Uncertainty Quantification](https://mlsanity.com/paper/2502.04406/) (2025)
- [Revisiting Heat Flux Analysis of Tungsten Monoblock Divertor on EAST using Physics-Informed Neural Network](https://mlsanity.com/paper/2508.03776/) (2025)
- [Evaluation and Verification of Physics-Informed Neural Models of the Grad-Shafranov Equation](https://mlsanity.com/paper/2504.21155/) (2025)
- [Towards Data-Efficient Cross-Device Generalization of Grad-Shafranov Equilibria via Transfer Learning Neural Operator](https://mlsanity.com/paper/2606.15512/) (2026)
- [Physics-Informed Neural Networks for Complex Eigenfrequency Identification and Mode Structure Reconstruction of the Ground-State ITG Branch](https://mlsanity.com/paper/2608.01850/) (2026)

## Abstract (arXiv)

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.