# Evaluation and Verification of Physics-Informed Neural Models of the Grad-Shafranov Equation

arXiv: https://arxiv.org/abs/2504.21155 · HTML: https://mlsanity.com/paper/2504.21155/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 31 references that are in the map.
- First posted on arXiv 29 Apr 2025.

## Builds on (most cited first, 14 of 14)

- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Adversarial Machine Learning at Scale](https://mlsanity.com/paper/1611.01236/) (2016)
- [Efficient Neural Network Robustness Certification with General Activation Functions](https://mlsanity.com/paper/1811.00866/) (2018)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [Fast and Complete: Enabling Complete Neural Network Verification with Rapid and Massively Parallel Incomplete Verifiers](https://mlsanity.com/paper/2011.13824/) (2020)
- [Efficient Formal Safety Analysis of Neural Networks](https://mlsanity.com/paper/1809.08098/) (2018)
- [General Cutting Planes for Bound-Propagation-Based Neural Network Verification](https://mlsanity.com/paper/2208.05740/) (2022)
- [Introduction to Neural Network Verification](https://mlsanity.com/paper/2109.10317/) (2021)
- [Neural Network Verification with Branch-and-Bound for General Nonlinearities](https://mlsanity.com/paper/2405.21063/) (2024)
- [Scaling Polyhedral Neural Network Verification on GPUs](https://mlsanity.com/paper/2007.10868/) (2020)
- [Deep neural network Grad-Shafranov solver constrained with measured magnetic signals](https://mlsanity.com/paper/1911.02882/) (2019)
- [Grad-Shafranov equilibria via data-free physics informed neural networks](https://mlsanity.com/paper/2311.13491/) (2023)
- [Investigating Guiding Information for Adaptive Collocation Point Sampling in PINNs](https://mlsanity.com/paper/2404.12282/) (2024)

## Abstract (arXiv)

Our contributions are motivated by fusion reactors that rely on maintaining magnetohydrodynamic (MHD) equilibrium, where the balance between plasma pressure and confining magnetic fields is required for stable operation. In axisymmetric tokamak reactors in particular, and under the assumption of toroidal symmetry, this equilibrium can be mathematically modelled using the Grad-Shafranov Equation (GSE). Recent works have demonstrated the potential of using Physics-Informed Neural Networks (PINNs) to model the GSE. Existing studies did not examine realistic scenarios in which a single network generalizes to a variety of boundary conditions. Addressing that limitation, we evaluate a PINN architecture that incorporates boundary points as network inputs. Additionally, we compare PINN model accuracy and inference speeds with a Fourier Neural Operator (FNO) model. Finding the PINN model to be the most performant, and accurate in our setting, we use the network verification tool Marabou to perform a range of verification tasks. Although we find some discrepancies between evaluations of the networks natively in PyTorch, compared to via Marabou, we are able to demonstrate useful and practical verification workflows. Our study is the first investigation of verification of such networks.