  Poisson's equation plays an important role in modeling many physical systems.
In electrostatic self-consistent low-temperature plasma (LTP) simulations,
Poisson's equation is solved at each simulation time step, which can amount to
a significant computational cost for the entire simulation. In this paper, we
describe the development of a generic machine-learned Poisson solver
specifically designed for the requirements of LTP simulations in complex 2D
reactor geometries on structured Cartesian grids. Here, the reactor geometries
can consist of inner electrodes and dielectric materials as often found in LTP
simulations. The approach leverages a hybrid CNN-transformer network
architecture in combination with a weighted multiterm loss function. We train
the network using highly-randomized synthetic data to ensure the
generalizability of the learned solver to unseen reactor geometries. The
results demonstrate that the learned solver is able to produce quantitatively
and qualitatively accurate solutions. Furthermore, it generalizes well on new
reactor geometries such as reference geometries found in the literature. To
increase the numerical accuracy of the solutions required in LTP simulations,
we employ a conventional iterative solver to refine the raw predictions,
especially to recover the high-frequency features not resolved by the initial
prediction. With this, the proposed learned Poisson solver provides the
required accuracy and is potentially faster than a pure GPU-based conventional
iterative solver. This opens up new possibilities for developing a generic and
high-performing learned Poisson solver for LTP systems in complex geometries.
