Fast neural Poincar\'e maps for toroidal magnetic fields

Poincar\'e maps for toroidal magnetic fields are routinely employed to study gross confinement properties in devices built to contain hot plasmas. In most practical applications, evaluating a Poincar\'e map requires numerical integration of a magnetic field line, a process that can be slow and that cannot be easily accelerated using parallel computations. We show that a novel neural network architecture, the H\'enonNet, is capable of accurately learning realistic Poincar\'e maps from observations of a conventional field-line-following algorithm. After training, such learned Poincar\'e maps evaluate much faster than the field-line integration method. Moreover, the H\'enonNet architecture exactly reproduces the primary physics constraint imposed on field-line Poincar\'e maps: flux preservation. This structure-preserving property is the consequence of each layer in a H\'enonNet being a symplectic map. We demonstrate empirically that a H\'enonNet can learn to mock the confinement properties of a large magnetic island by using coiled hyperbolic invariant manifolds to produce a sticky chaotic region at the desired island location. This suggests a novel approach to designing magnetic fields with good confinement properties that may be more flexible than ensuring confinement using KAM tori.