Poisson CNN: Convolutional Neural Networks for the Solution of the Poisson Equation with Varying Meshes and Dirichlet Boundary Conditions

The Poisson equation is commonly encountered in engineering, including in computational fluid dynamics where it is needed to compute corrections to the pressure field. We propose a novel fully convolutional neural network (CNN) architecture to infer the solution of the Poisson equation on a 2D Cartesian grid of varying size and spacing given the right hand side term, arbitrary Dirichlet boundary conditions and grid parameters which provides unprecendented versatility in this application. 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. Analytical test cases indicate that our CNN architecture is capable of predicting the correct solution of a Poisson problem with mean percentage errors of 15% and promises improvements in wall-clock runtimes for large problems. Furthermore, even when predicting on meshes denser than previously encountered, our model demonstrates encouraging capacity to reproduce the correct solution profile.