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

TL;DR

This paper tackles real-time plasma equilibrium modeling by solving the Grad-Shafranov equation with physics-informed neural networks (PINNs) that take boundary geometry as explicit inputs, and compares them to Fourier Neural Operators (FNOs). It introduces a dataset and training pipeline based on Solov'ev-reduced GSE with arbitrary 2D boundaries, along with adaptive resampling and PDE/boundary losses to enforce physics. The authors demonstrate that PINNs achieve competitive accuracy and substantially faster inference than FNOs in this setting, while also establishing practical verification workflows using Marabou to assess robustness and detect inconsistencies under boundary extrapolation. The work shows that formal verification is feasible for PINNs in plasma physics, enabling error detection and reliability checks essential for safety-critical tokamak control scenarios, and highlights both the potential and limitations of verification under finite-precision computation. Overall, the study advances boundary-generalization and verifiable physics-informed learning for tokamak equilibrium modeling, with implications for real-time diagnostics and control.”

Abstract

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.

Figures (4)

• Figure 1: Example of adaptive resampling visualization: Green dots represent boundary points, blue dots indicate randomly generated points, and orange dots highlight points with higher errors that will be prioritized for the next iteration.
• Figure 2: Diagram illustrating the network used for evaluation, showcasing the PINN and FNO architectures.
• Figure 3: An overview of the process flow for model verification using Marabou. (a) illustrates the process of converting the model from PyTorch to ONNX, where the shape of the input sample is required. (b) shows the verification flow, where Marabou, given the ONNX model and queries, produces the result through Gurobi.
• Figure 4: Predicted flux surface for $\kappa$ = 4.5 boundary vs the true boundary (blue dots).