Fractional Fractal Feature Learning and Multifractal Scaling Laws in Artificial Neural Networks

Fractal geometry provides a powerful mathematical framework for describing irregular and self-similar structures that arise in many natural and engineered systems. At the same time, artificial intelligence has emerged as an effective tool for extracting patterns from high-dimensional data. Despite these advances, existing machine learning models often lack explicit mechanisms for capturing fractal and multifractal scaling laws present in complex datasets. This paper introduces a unified mathematical and computational framework that integrates fractal geometry, fractional calculus, and artificial intelligence for the analysis of multiscale complex structures. Inspired by fractal neural architectures, we propose a new class of fractional fractal descriptors capable of capturing higher-order spatial irregularities. These descriptors extend classical fractal measures such as fractal dimension, lacunarity, and percolation probability by incorporating fractional scaling operators. A neural learning architecture is further developed to estimate fractal and multifractal parameters directly from spatial data using multiscale fractal feature representations. Theoretical analysis establishes new scaling laws describing the propagation of fractal features across neural network layers and provides stability conditions for the preservation of fractal structures in deep models. Extensive numerical experiments using synthetic fractal datasets demonstrate that the proposed framework provides accurate estimation of fractal parameters and significantly improves multiscale pattern recognition compared with classical convolutional neural network approaches. The proposed methodology opens new directions for combining fractal mathematics with artificial intelligence in the analysis of complex systems arising in physics, biology, finance, and engineering.