Neural Network Operator-Based Fractal Approximation: Smoothness Preservation and Convergence Analysis

TL;DR

This paper introduces a neural network operator-based method for constructing fractal interpolation functions that preserve smoothness and converge to the original function, advancing approximation theory with neural network techniques.

Contribution

It develops a novel neural network operator-based approach for creating smoothness-preserving fractal interpolation functions with convergence guarantees.

Findings

Fractal functions constructed using neural network operators preserve the smoothness of original functions.

The method guarantees convergence of the fractal functions to the original function under certain conditions.

Provides uniform approximation error bounds using approximation theory tools.

Abstract

This paper presents a new approach of constructing $\alpha$-fractal interpolation functions (FIFs) using neural network operators, integrating concepts from approximation theory. Initially, we construct $\alpha$-fractals utilizing neural network-based operators, providing an approach to generating fractal functions with interpolation properties. Based on the same foundation, we have developed fractal interpolation functions that utilize only the values of the original function at the nodes or partition points, unlike traditional methods that rely on the entire original function. Further, we have constructed \(\alpha\)-fractals that preserve the smoothness of functions under certain constraints by employing a four-layered neural network operator, ensuring that if \(f \in C^{r}[a,b]\), then the corresponding fractal \(f^{\alpha} \in C^{r}[a,b]\). Furthermore, we analyze the convergence of these $\alpha$-fractals to the original function under suitable conditions. The work uses key approximation theory tools, such as the modulus of continuity and interpolation operators, to develop convergence results and uniform approximation error bounds.