Level Set Learning for Poincaré Plots of Symplectic Maps

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.