  Reduced order models are becoming increasingly important for rendering
complex and multiscale spatio-temporal dynamics computationally tractable. The
computational efficiency of such surrogate models is especially important for
design, exhaustive exploration and physical understanding. Plasma simulations,
in particular those applied to the study of ${\bf E}\times {\bf B}$ plasma
discharges and technologies, such as Hall thrusters, require substantial
computational resources in order to resolve the multidimentional dynamics that
span across wide spatial and temporal scales. Although high-fidelity
computational tools are available to simulate such systems over limited
conditions and in highly simplified geometries, simulations of full-size
systems and/or extensive parametric studies over many geometric configurations
and under different physical conditions are computationally intractable with
conventional numerical tools. Thus, scientific studies and industrially
oriented modeling of plasma systems, including the important ${\bf E}\times
{\bf B}$ technologies, stand to significantly benefit from reduced order
modeling algorithms. We develop a model reduction scheme based upon a {\em
Shallow REcurrent Decoder} (SHRED) architecture. The scheme uses a neural
network for encoding limited sensor measurements in time (sequence-to-sequence
encoding) to full state-space reconstructions via a decoder network. Based upon
the theory of separation of variables, the SHRED architecture is capable of (i)
reconstructing full spatio-temporal fields with as little as three point
sensors, even the fields that are not measured with sensor feeds but that are
in dynamic coupling with the measured field, and (ii) forecasting the future
state of the system using neural network roll-outs from the trained time
encoding model.
