Constructing a class of neural network operators approximating integrable functions
TL;DR
A constructive technique is employed to develop a broad and novel class of FNNs, proving the direct, inverse, and approximation characteristic equivalence theorems for Lebesgue integrable functions.
Abstract
Directly constructing feedforward neural networks (FNNs) as approximators and establishing their approximation theorems is extremely important. This paper employs a constructive technique to develop a broad and novel class of FNNs, proving the direct, inverse, and approximation characteristic equivalence theorems for Lebesgue integrable functions. This study's primary innovations are: The proposed technique applies to a wide range of sigmoidal functions that form the activation function of the FNNs, including not just the conventional logistic and tangent functions; The established direct theorem of approximation reveals that the constructed FNN avoids the demand for training cycles and subsequent generalization testing, as well as avoiding local minimum in optimization;The obtained equivalence characterization theorem of approximation should be useful for designing FNN structures based on their approximation capacity. These theorems are built using K-functionals, smoothness moduli, and the Berens-Lorentz lemma, as well as Jackson-and-Bernstein type inequalities. Some numerical examples support the efficiency of the theoretical results.