  We present a method to solve initial and boundary value problems using
artificial neural networks. A trial solution of the differential equation is
written as a sum of two parts. The first part satisfies the boundary (or
initial) conditions and contains no adjustable parameters. The second part is
constructed so as not to affect the boundary conditions. This part involves a
feedforward neural network, containing adjustable parameters (the weights).
Hence by construction the boundary conditions are satisfied and the network is
trained to satisfy the differential equation. The applicability of this
approach ranges from single ODE's, to systems of coupled ODE's and also to
PDE's. In this article we illustrate the method by solving a variety of model
problems and present comparisons with finite elements for several cases of
partial differential equations.
