Constructing a class of neural network operators approximating integrable functions

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.