Evaluation and Verification of Physics-Informed Neural Models of the Grad-Shafranov Equation
This work addresses the problem of verifying neural models for magnetohydrodynamic equilibrium in fusion reactors, which is crucial for stable operation, but it is incremental as it builds on existing PINN methods by adding verification workflows.
The authors tackled the challenge of modeling the Grad-Shafranov Equation for fusion reactors using Physics-Informed Neural Networks (PINNs), evaluating a PINN architecture that generalizes to various boundary conditions and comparing it to a Fourier Neural Operator model, with the PINN achieving the best performance and accuracy in their setting, and they demonstrated the first verification of such networks using Marabou, though noting some discrepancies in evaluations.
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.