Richards's curve induced Banach space valued multivariate neural network approximation.

Area of Science:

• Numerical Analysis
• Functional Analysis
• Machine Learning

Background:

• Approximation theory deals with approximating complex functions using simpler ones.
• Neural networks are increasingly used in approximation tasks.
• Banach spaces provide a framework for studying function spaces.

Purpose of the Study:

• To develop and analyze new multivariate neural network operators for function approximation.
• To investigate the approximation capabilities of normalized, quasi-interpolation, Kantorovich-type, and quadrature-type operators.
• To examine the effect of iterated operators on approximation quality.

Main Methods:

• Utilizing multivariate normalized, quasi-interpolation, Kantorovich-type, and quadrature-type neural network operators.
• Establishing multidimensional Jackson type inequalities.
• Employing a multidimensional density function based on the Richards's curve (a generalized logistic function).
• Analyzing pointwise and uniform approximation errors.

Main Results:

• Quantitative approximations of continuous multivariate functions in Banach spaces were achieved.
• The effectiveness of iterated operators in enhancing approximation was demonstrated.
• Multidimensional Jackson type inequalities provided error bounds based on continuity moduli and derivatives.
• The feed-forward neural network architecture involved one hidden layer.

Conclusions:

• The proposed neural network operators provide effective tools for approximating continuous multivariate functions.
• The theoretical framework, based on Jackson type inequalities, validates the approximation quality.
• The use of Richards's curve-based density functions offers a novel approach in this domain.