AI 中文总结
该研究提出几何条件傅里叶神经算子（FNO）作为代理模型，可毫秒级高精度求解双零自由边界Grad-Shafranov平衡，计算速度较FreeGS提升数十至数百倍，适用于磁约束聚变相关工作流程。
AI 中文摘要
Grad-Shafranov（GS）方程描述了托卡马克等离子体中的理想磁流体动力学平衡，自由边界GS求解器是偏滤器平衡建模的核心，但非线性皮卡迭代会带来计算成本和样本相关的延迟，在优化、建模和控制导向的循环中可能变得难以承受。本文训练了一个几何条件傅里叶神经算子（FNO），用于学习从空间坐标、标量运行参数（中心压强P_axis、等离子体电流I_p、真空磁场f_vac）以及规定的X点位置到极向通量场ψ(R,Z)的约束正向映射。该模型在由FreeGS生成的受控双零自由边界平衡族上进行训练，针对单一固定装置几何结构和规定拓扑。最优模型的平均相对L²误差为0.05%，在训练样本数N_train∈{500,1000,2000,5000}范围内，测试误差遵循经验幂律N^-0.68；它能将X点恢复到0.2厘米以内，将O点定位到0.03厘米。作为物理一致性诊断，预测场满足外部有限差分GS残差评估，其水平与真实场相当，平均归一化残差为2.29，与使用相同诊断的FreeGS基线的2.29±0.06无差异。训练后的FNO在GPU上求解一个平衡耗时2.77毫秒，在CPU上耗时25.6毫秒，相对于此处配置的FreeGS分别实现了约640倍和约69倍的加速，且延迟接近确定性（p95/中位数=1.01）。这些结果表明，神经算子代理模型可在规定拓扑和装置几何结构内，为磁约束聚变工作流程提供准确、几何精度高的毫秒级平衡评估。
英文摘要
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 $ψ(R,Z)$. 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.
Comments13 pages, 8 figures, 3 tables
