Interpolation for neural-network operators activated with a generalized logistic-type function

Abstract This paper defines a family of neural-network interpolation operators. The first derivative of generalized logistic-type functions is considered as a density function. Using the first-order uniform approximation theorem for continuous functions defined on the finite intervals, the interpolation properties of these operators are presented. A Kantorovich-type variant of the operators $F_{n}^{a,\varepsilon} $ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>F</mml:mi> <mml:mi>n</mml:mi> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo>,</mml:mo> <mml:mi>ε</mml:mi> </mml:mrow> </mml:msubsup> </mml:math> is also introduced. The approximation of Kantorovich-type operators in $L_{P}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>P</mml:mi> </mml:msub> </mml:math> spaces with $1 \leq p\leq \infty $ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>1</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>p</mml:mi> <mml:mo>≤</mml:mo> <mml:mi>∞</mml:mi> </mml:math> is studied. Further, different combinations of the parameters of our generalized logistic-type activation function $\theta _{s, a}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>θ</mml:mi> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>,</mml:mo> <mml:mi>a</mml:mi> </mml:mrow> </mml:msub> </mml:math> are examined to see which parameter values might give us a more efficient activation function. By choosing suitable parameters for the operator $F_{n}^{a,\varepsilon} $ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>F</mml:mi> <mml:mi>n</mml:mi> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo>,</mml:mo> <mml:mi>ε</mml:mi> </mml:mrow> </mml:msubsup> </mml:math> and the Kantorovich variant of the operator $F_{n}^{a,\varepsilon} $ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>F</mml:mi> <mml:mi>n</mml:mi> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo>,</mml:mo> <mml:mi>ε</mml:mi> </mml:mrow> </mml:msubsup> </mml:math> , the approximation of various function examples is studied.