Millisecond-Scale Neural Operator Surrogates for Double-Null Free-Boundary Grad-Shafranov Equilibria

The Grad-Shafranov (GS) equation governs ideal magnetohydrodynamic equilibrium in tokamak plasmas. Free-boundary GS solvers are central to diverted-equilibrium modeling, but nonlinear Picard iteration introduces computational cost and sample-dependent latency that can become prohibitive in optimization, modeling, and control-oriented loops. Here we train a geometrically conditioned Fourier Neural Operator (FNO) to learn a constrained forward map from spatial coordinates, scalar operating parameters $(P_{\mathrm{axis}}, I_p, f_{\mathrm{vac}})$, and prescribed X-point locations to the poloidal-flux field $\psi(R,Z)$. The model is trained on a controlled family of constrained double-null free-boundary equilibria generated with \textsc{FreeGS} for a single fixed machine geometry and prescribed topology. The best model achieves a mean relative $L^2$ error of $0.05\%$, with test error following an empirical $N^{-0.68}$ power law over $N_{\mathrm{train}}\in\{500,1000,2000,5000\}$. It recovers both X-points to within $0.2$ cm and localizes the O-point to $0.03$ cm. As a physics-consistency diagnostic, the predicted fields satisfy an external finite-difference GS residual evaluation at the same level as the ground-truth fields, with mean normalized residual $2.29$, indistinguishable from the $2.29\pm0.06$ \textsc{FreeGS} baseline using the same diagnostic. The trained FNO evaluates one equilibrium in $2.77$ ms on GPU and $25.6$ ms on CPU, corresponding to speedups of ${\sim}640\times$ and ${\sim}69\times$ relative to \textsc{FreeGS} as configured here, with near-deterministic latency (p95/median $=1.01$). These results show that neural-operator surrogates can provide accurate, geometrically precise, millisecond-scale equilibrium evaluations for magnetic-confinement fusion workflows within a prescribed topology and machine geometry.