Approximation by network operators with logistic activation functions

Zhixiang Chen, Feilong Cao, Jinjie Hu

Abstract: This paper aims to study the construction and multivariate approximation of a class of network operators with logistic sigmoidal functions. First, a class of even and bell-shaped function with support on R<math><mrow is="true"><mi mathvariant="double-struck" is="true">R</mi></mrow></math> is constructed by using appropriate translation and combination of the logistic function. Then, the constructed function is employed as activation function to construct a kind of so-called Cardaliaguet–Euvrard type network operators. Finally, these network operators are used to approximate bivariate functions in C[-1,1]2<math><mrow is="true"><msub is="true"><mrow is="true"><mi is="true">C</mi></mrow><mrow is="true"><msup is="true"><mrow is="true"><mo stretchy="false" is="true">[</mo><mo is="true">-</mo><mn is="true">1</mn><mtext is="true">,</mtext><mn is="true">1</mn><mo stretchy="false" is="true">]</mo></mrow><mrow is="true"><mn is="true">2</mn></mrow></msup></mrow></msub></mrow></math>, and a Jackson type theorem for the approximation errors is established.