Constructive function-approximation by three-layer artificial neural networks

Constructive theorems of three-layer artificial neural networks with (1) trigonometric, (2) piecewise linear, and (3) sigmoidal hidden-layer units are proved in this paper. These networks approximate 2 pi-periodic pth-order Lebesgue-integrable functions (L-2 pi(p)) on R-m to R-n for p greater than or equal to 1 with L-2 pi(p) - norm. (In the case of (1), the networks also approximate 2 pi-periodic continuous functions (C-2 pi) with C-2 pi-norm.) These theorems provide explicit equational representations of these approximating networks, specifications for their numbers of hidden-layer units, and explicit formulations of their approximation-error estimations. The function-approximating networks and the estimations of their approximation errors can practically and easily be calculated from the results. The theorems can easily be applied to the approximation of a nonperiodic function defined in a bounded set on R-m to R-n. (C) 1998 Elsevier Science Ltd. All rights reserved.