AI-GUIDED FRACTIONAL-FRACTAL SCALING LAWS AND MULTIFRACTAL EVOLUTION UNDER FRACTIONAL DYNAMICS

Abstract

We develop a unified mathematical and computational framework linking fractional calculus, fractal geometry, scaling laws, and artificial intelligence (AI). We establish a universal scaling-shift theorem demonstrating that fractional operators systematically transform fractal and multifractal spectra of self-affine stochastic processes. For any self-affine process with Hurst exponent H, we prove that a fractional derivative of order beta produces a new process with scaling exponent H-beta = H - beta, implying a fractal dimension shift D-beta = D + beta. We further derive a multifractal spectral deformation law tau(beta)(q) = tau(q) - q beta, which demonstrates how fractional dynamics rearranges local singularities. To connect theory with observations, we present an AI-facilitated framework of neural operators that can compute fractal dimensions and detect fractional orders from observations. Numerical experiments confirm the theoretical predictions and demonstrate strong agreement between analytical scaling laws and AI-inferred parameters.