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Revision with unchanged content. This work presents a mesh-free method for solving BVPs whose key to success is incorporating knowledge the given boundary conditions into the appro ximate 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 solu tion even before optimization begins, significantly simplifying the optimi za tion process after it has begun.

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