APPROXIMATIONS OF CONTINUOUS FUNCTIONALS BY NEURAL NETWORKS WITH APPLICATION TO DYNAMIC-SYSTEMS

The main concern of this paper is to give several strong results on neural network representation in an explicit form. Under very mild conditions a functional defined on a compact set in C[a, b] or L(P)[a, b], spaces of infinite dimensions, can be approximated arbitrarily well by a neural network with one hidden layer. In particular, if U is a compact set in C[a, b], sigma is a bounded sigmoidal function, and S is a continuous functional defined on U, then for all u is an element of U, flu) can be approximated by Sigma(i=1)(N)c(i) sigma)(Sigma(i=0)(m) xi(i,j)u(x(j))+theta(i)) where c(i), xi(ij), theta(i) are real numbers. u(x(j)) is the value of u evaluated at point x(j). These results are a significant development beyond earlier works, where theorems of approximating continuous functions defined on R(n), a space of finite dimension by neural networks with one hidden layer, were given. Finally, all the results are shown applicable to the approximation of the output of dynamic systems at any particular time.