  Managing divertor plasmas is crucial for operating reactor scale tokamak
devices due to heat and particle flux constraints on the divertor target.
Simulation is an important tool to understand and control these plasmas,
however, for real-time applications or exhaustive parameter scans only simple
approximations are currently fast enough. We address this lack of fast
simulators using neural PDE surrogates, data-driven neural network-based
surrogate models trained using solutions generated with a classical numerical
method. The surrogate approximates a time-stepping operator that evolves the
full spatial solution of a reference physics-based model over time. We use
DIV1D, a 1D dynamic model of the divertor plasma, as reference model to
generate data. DIV1D's domain covers a 1D heat flux tube from the X-point
(upstream) to the target. We simulate a realistic TCV divertor plasma with
dynamics induced by upstream density ramps and provide an exploratory outlook
towards fast transients. State-of-the-art neural PDE surrogates are evaluated
in a common framework and extended for properties of the DIV1D data. We
evaluate (1) the speed-accuracy trade-off; (2) recreating non-linear behavior;
(3) data efficiency; and (4) parameter inter- and extrapolation. Once trained,
neural PDE surrogates can faithfully approximate DIV1D's divertor plasma
dynamics at sub real-time computation speeds: In the proposed configuration,
2ms of plasma dynamics can be computed in $\approx$0.63ms of wall-clock time,
several orders of magnitude faster than DIV1D.
