Approximation by a class of neural network operators on scattered data

Abstract

Scattered data are a class of common data in the real world. Naturally, how to efficiently process scattered data is important. This paper uses a class of feedforward neural networks with four layers as tool to fit scattered data and establishes the estimates of the approximation error. In particular, an inverse theorem of the approximation is established. Concretely, we first extend an existed result on false[−1,1false]2$$ {\left[-1,1\right]}^2 $$ to the case of arbitrary bounded convex set boldΩ$$ \boldsymbol{\Omega} $$ in ℝd$$ {\mathbb{R}}^d $$. Secondly, we introduce a modified feedforward neural network with four layers, which is a class of quasi‐interpolation operators and can keep the smoothness of the objective function. By establishing two Bernstein‐type inequalities for the operators, we establish both the direct and converse results of the approximation by the operator, which follows the equivalence characterization theorem of the approximation.