  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.
