**
Numerical Solution of a Calculus of Variations Problem Using the Feedforward Neural Network Architecture
,**

Andrew J. Meade, Jr. and Hans C. Sonneborn,

Submitted to *Computing Systems in Engineering*, 1995.

**Keywords:**:
feedforward artificial neural networks, neural
computation, calculus of variations.

**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.

This work was supported under NASA grant NAG 1-1433.