This work presents a mesh-free method for solving BVPs whose key to success
is incorporating knowledge the given boundary conditions into the approximate
solution to the desired differential equation. This method generates
an approximate solution continuous over the problem domain of arbitrary
shape, and the approximate solution exactly satisfies all boundary conditions
whether Dirichlet and/or Neumann. The approximate solution is thus exact in
either value or slope everywhere along the boundary, greatly simplifying the
effort required by the artificial neural network algorithm, which optimizes the
approximate solution for the interior of the domain. This method builds
boundary information directly into the form of the approximate solution
rather than simply using boundary value information to define a system of
equations for solution as in the finite-element method. The result is an
approximate solution which can be startlingly similar to the analytical solution
even before optimization begins, significantly simplifying the optimization
process after it has begun.