文档介绍：Advances in EngineeringSoftware 21 (1996) 2 13-225
Copyright Q 1996 Elsevier Science Limited
Printed in Great Britain. All rights reserved
PII:S50965-9978(96)00029-4 096559978/96/$
ELSEVIER
Numerical solution of a calculus of variations
problem using the feedforward neural
network architecture
Andrew J. Meade Jr. & Hans C. Sonneborn
Department of Mechanical Engineering and Materials Science, William Marsh Rice University, Houston, TX -1892, USA
(Received for publication 6 June 1996)
It is demonstrated, through theory and numerical example, how it is possible to
construct directly and noniteratively a feedforward work 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 L2 norm
of work approximation error decreasesquadratically 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. Copyright 0 1996 Elsevier
Science Limited.
Kev words: Feedforward artificial works, putation, calculus
of variations.
A F
NOMENCLATURE (Yj >. . !O!j input weights used for construction of
the ith spline
Symbols Explanation e:,...,e; bias weights used for construction of the
ith spline
X independent variable N number of basis functions used in
Y dependent variable solution approximation
T total number of hidden layer nodes Ulj,. . output weights used in construction of
r4 transfer function of the qth hidden layer