Fast Poincar\'e maps for magnetic fields using symplectic neural networks

Field-line Poincar\'e maps are powerful tools for analyzing the global behavior of magnetic fields in magnetic fusion devices. In some special cases, the Poincar\'e map may be derived by hand, but in most practical applications the map is approximated by numerically integrating the streamlines for the magnetic field. We present a new method for computing approximate Poincar\'e maps based on a novel neural network architecture called the H\'enon Network. A H\'enon Network is trained in a supervised fashion by showing it results from fourth-order Runge-Kutta field-line following simulations. After training, the network's input-to-output mapping gives an exactly-flux-conserving (i.e. symplectic) approximation of the Poincar\'e map. Moreover, evaluating such a neural approximation of the Poincar\'e map is orders of magnitude faster than evaluating an approximation based on field-line following.