Numerical solution of a calculus of variations problem using the feedforward neural network architecture

Abstract

It is demonstrated, through theory and numerical example, how it is possible to construct directly and noniteratively a feedforward neural network to solve a calculus of variations problem. The method, using the piecewise linear and cubic sigmoid transfer functions, is linear in storage and processing time. The L 2 norm of the network approximation error decreases quadratically with the piecewise linear transfer function and quartically with the piecewise cubic sigmoid as the number of hidden layer neurons increases. The construction requires imposing certain constraints on the values of the input, bias, and output weights, and the attribution of certain roles to each of these parameters. All results presented used the piecewise linear and cubic sigmoid transfer functions. However, the noniterative approach should also be applicable to the use of hyperbolic tangents and radial basis functions.