  The Poisson equation is commonly encountered in engineering, for instance in
computational fluid dynamics (CFD) where it is needed to compute corrections to
the pressure field to ensure the incompressibility of the velocity field. In
the present work, we propose a novel fully convolutional neural network (CNN)
architecture to infer the solution of the Poisson equation on a 2D Cartesian
grid with different resolutions given the right hand side term, arbitrary
boundary conditions and grid parameters. It provides unprecedented versatility
for a CNN approach dealing with partial differential equations. The boundary
conditions are handled using a novel approach by decomposing the original
Poisson problem into a homogeneous Poisson problem plus four inhomogeneous
Laplace sub-problems. The model is trained using a novel loss function
approximating the continuous $L^p$ norm between the prediction and the target.
Even when predicting on grids denser than previously encountered, our model
demonstrates encouraging capacity to reproduce the correct solution profile.
The proposed model, which outperforms well-known neural network models, can be
included in a CFD solver to help with solving the Poisson equation. Analytical
test cases indicate that our CNN architecture is capable of predicting the
correct solution of a Poisson problem with mean percentage errors below 10%, an
improvement by comparison to the first step of conventional iterative methods.
Predictions from our model, used as the initial guess to iterative algorithms
like Multigrid, can reduce the RMS error after a single iteration by more than
90% compared to a zero initial guess.
