Accelerating Poisson equation solvers with physics informed neural networks

Poisson equations are a class of partial differential equations (PDEs) encountered in several areas of physics, from fluid mechanics to electrostatics. Iterative solvers, such as multigrid and conjugate gradient algorithms, are commonly employed to solve Poisson equations but become computationally expensive and require an increasing number of iterations for large problems. In this work, we focus on the pressure equation encountered within the pressure-velocity coupling for incompressible flows, which ensures a divergence-free velocity field. We train physics-informed neural networks (PINNs) to solve the pressure equation, taking spatial coordinates and source terms as model inputs. Our model is easily parallelised and agnostic to mesh resolution, allowing us to train on data from coarse grids and predict on resolved cases in parallel. We deploy the trained Tensorflow model within the pimpleFoam solver in OpenFOAM and observe a reduction in pressure solver iterations when using the neural network predictions as an initial guess for the linear solver compared to using the previous pressure field. We then study the performance impact of using PINNs as a first guess and investigate optimal conditions for their use with linear solvers.