  In this paper we use deep feedforward artificial neural networks to
approximate solutions to partial differential equations in complex geometries.
We show how to modify the backpropagation algorithm to compute the partial
derivatives of the network output with respect to the space variables which is
needed to approximate the differential operator. The method is based on an
ansatz for the solution which requires nothing but feedforward neural networks
and an unconstrained gradient based optimization method such as gradient
descent or a quasi-Newton method.
  We show an example where classical mesh based methods cannot be used and
neural networks can be seen as an attractive alternative. Finally, we highlight
the benefits of deep compared to shallow neural networks and device some other
convergence enhancing techniques.
