Rates of approximation by neural network interpolation operators

Highlights：

• In the present paper, we introduce a general class of activation functions that are linear combinations of general sigmoidal functions. By using the new introduced activation functions, we construct a single layer neural network interpolation operator (Eq. (1.9)). The following properties of the operator are investigated: the rate of approximation by the operator for continuous functions; some important inequalities of the derivative of the operator; the converse result of approximation; a special combinations of the operators, which can approximate the object function and its derivative simultaneously; both the direct and converse theorem of approximation by Kantorovich variant of the operator. We also give some examples of the newly defined activation functions. Finally, we give some numerical examples to demonstrate the validity of the obtained results.