Constructive function-approximation by three-layer artificial neural networks.
basic_science · Level V
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Abstract
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 2pi-periodic pth-order Lebesgue-integrable functions (L(p)(2pi)) on R(m) to R(n) for p>/=1 with L(p)(2pi)-norm. (In the case of (1), the networks also approximate 2pi-periodic continuous functions (C(2pi)) with C(2pi)-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 non-periodic function defined in a bounded set on R(m) to R(n).