  Can neural networks learn to solve partial differential equations (PDEs)? We
investigate this question for two (systems of) PDEs, namely, the Poisson
equation and the steady Navier--Stokes equations. The contributions of this
paper are five-fold. (1) Numerical experiments show that small neural networks
(< 500 learnable parameters) are able to accurately learn complex solutions for
systems of partial differential equations. (2) It investigates the influence of
random weight initialization on the quality of the neural network approximate
solution and demonstrates how one can take advantage of this non-determinism
using ensemble learning. (3) It investigates the suitability of the loss
function used in this work. (4) It studies the benefits and drawbacks of
solving (systems of) PDEs with neural networks compared to classical numerical
methods. (5) It proposes an exhaustive list of possible directions of future
work.
