Many important qualities of plasma confinement devices can be determined via the Poincar\'e plot of a symplectic return map. These qualities include the locations of periodic orbits, magnetic islands, and chaotic regions of phase space. However, every evaluation of the magnetic return map requires solving an ODE, meaning a detailed Poincar\'e plot can be expensive to create. Here, we propose a kernel-based method of learning a single labeling function that is approximately invariant under the symplectic map. From the labeling function, we can recover the locations of invariant circles, islands, and chaos with few evaluations of the underlying symplectic map. Additionally, the labeling function comes with a residual, which serves as a measure of how invariant the label function is, and therefore as an indirect measure of chaos and map complexity.

翻译：等离子体约束装置的许多重要特性可通过辛回归映射的庞加莱图来确定，这些特性包括周期轨道位置、磁岛以及相空间混沌区域。然而，每次对磁回归映射的评估都需要求解常微分方程，这意味着绘制详细的庞加莱图可能耗费高昂。本文提出一种基于核函数的单一标定函数学习方法，该函数在辛映射下近似保持不变。通过该标定函数，我们可在仅需少量底层辛映射评估的情况下恢复不变圆、岛状结构及混沌区域的位置。此外，该标定函数附带残差项，可衡量标定函数的不变性程度，从而间接反映混沌程度与映射复杂度。