A Physics-Informed Neural Network for Fast Equilibrium Calculations in Tokamaks

F. Galfrascoli, Z. Wang, T. Rafiq, E. Schuster

31st IEEE Symposium on Fusion Energy (SOFE)

Cambridge, MA, USA, June 23-26, 2025

Tokamak operation relies on accurate and efficient free-boundary equilibrium (FBE) solvers. Traditional solution methods, such as those based on finite-difference and Picard iteration [1], can be computationally expensive. This work introduces a novel approach using physics-informed neural networks (PINNs) to solve both direct and inverse FBE problems. The direct problem determines the 2D poloidal-flux map given plasma conditions and poloidal-field (PF) coil currents. The inverse problem calculates the required PF coil currents and the 2D poloidal-flux map for specified plasma conditions and target shapes. The PINN surrogate models, first developed to address the direct problem [2] and now to address the inverse problem, employ fully connected multi-layer perceptron (MLP) architectures and incorporate the Grad-Shafranov equation as a physical constraint in their loss functions, ensuring consistency between plasma conditions, PF coil currents, and the resulting 2D poloidal-flux map. Extensive training datasets were generated to encompass a broad range of plasma configurations, including variations in plasma current, poloidal beta, major and minor radii, triangularity, and elongation. The performance of the inverse-mode PINN developed in this work was benchmarked against conventional FBE solvers, demonstrating high accuracy in replicating the outputs of the FBE solver and achieving a low root-mean-squared error (RMSE) for the magnetic flux maps. Furthermore, the direct and inverse-mode PINN models were integrated into a unified solver capable of handling both types of equilibrium problems. This unified framework significantly enhances computational efficiency, completing calculations in a fraction of a second compared to tens of seconds for traditional iterative solvers. By combining speed and accuracy, the integrated PINN-based solver offers a robust tool for real-time control and optimization applications in tokamak research.

[1] Song, X. et al. (2024). Plasma, 7(4), 842-857.
[2] Wang, Z. et al. (2024). IEEE Transactions on Plasma Science, 52(9), 4147-4153.

*Supported by the US DOE (DE-SC0010537).