Neural network interpolation operators optimized by Lagrange polynomial. (September 2022)

• Record Type: Journal Article
• Authors: Wang, Guoshun; Yu, Dansheng; Zhou, Ping
• Is Part Of: Neural networks. Volume 153(2022)
• Page Start: 179
• Page End: 191
• Publication Date: 2022-09
• Subjects: Sigmoidal function -- Neural network operators -- Interpolation -- Uniform approximate; Neural computers -- Periodicals; Neural networks (Computer science) -- Periodicals; Neural networks (Neurobiology) -- Periodicals; Nervous System -- Periodicals; Ordinateurs neuronaux -- Périodiques; Réseaux neuronaux (Informatique) -- Périodiques; Réseaux neuronaux (Neurobiologie) -- Périodiques; Neural computers; Neural networks (Computer science); Neural networks (Neurobiology); Periodicals; 006.32
• DOI: 10.1016/j.neunet.2022.06.007
• Languages: English
• ISSNs: 0893-6080

Abstract

Abstract: In this paper, we introduce a new type of interpolation operators by using Lagrange polynomials of degree r, which can be regarded as feedforward neural networks with four layers. The approximation rate of the new operators can be estimated by the ( r + 1 ) -th modulus of smoothness of the objective functions. By adding some smooth assumptions on the activation function, we establish two important inequalities of the derivatives of the operators. With these two inequalities, by using the K -functional and Berens–Lorentz lemma in approximation theory, we establish the converse theorem of approximation. We also give the Voronovskaja-type asymptotic estimation of the operators for smooth functions. Furthermore, we extend our operators to the multivariate case, and investigate their approximation properties for multivariate functions. Finally, some numerical examples are given to demonstrate the validity of the theoretical results obtained and the superiority of the operators.