Time-delay neural networks, Volterra series, and rates of approximation (Q1273815)

Problems involving the approximation of nonlinear input-output maps arise often in engineering and science. The author considers a large family of approximately-finite-memory causal time-invariant maps \(G\) from an input set \(S\) to a set of \({\mathbb R}\)-valued functions, with members of both sets of functions defined on the nonnegative integers, and he gives an upper bound on the error in approximating a \(G\) using a two-stage structure consisting of a tapped delay line followed by a static neural network. As an application, information is given concerning the long-standing problem of determining the order of a Volterra-series approximation so that a given quality of approximation can be achieved. Corresponding results are also given for the approximation of not necessarily causal input-output maps with inputs and outputs that may depend on more than one variable. These results are of interest, for example, in connection with image processing.