用神经网络构造保持光滑性的分形插值函数，实现精准逼近。
Neural Network Operator-Based Fractal Approximation: Smoothness Preservation and Convergence Analysis
- 用四层神经网络构造分形插值函数，仅需节点值即可生成。
- 在特定条件下，原函数光滑则对应分形函数也光滑，保持C^r连续性。
- 理论证明分形函数可一致收敛于原函数，误差有明确上界。
本文提出一种基于神经网络算子的新型α-分形插值函数构建方法，融合逼近论思想。首先，利用神经网络构造α-分形，实现具有插值性质的分形函数生成；其次，所提方法仅需原函数在节点处的取值，无需整体信息，区别于传统方法。进一步，通过四层神经网络算子，在特定约束下实现了分形函数对原函数光滑性的保持：若f ∈ C^r[a,b]，则对应的分形函数f^α ∈ C^r[a,b]。同时，在合适条件下分析了α-分形对原函数的收敛性，并借助模连续性和插值算子等逼近论工具，建立了统一逼近误差的上界。该研究为分形插值提供了新的可计算、可分析且保持结构特性的方法。
原文摘要 · Abstract (English)
This paper presents a new approach of constructing $α$-fractal interpolation functions (FIFs) using neural network operators, integrating concepts from approximation theory. Initially, we construct $α$-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 \(α\)-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 \(f^α \in C^{r}[a,b]\). Furthermore, we analyze the convergence of these $α$-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.
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