Advancements in Fractional Neural Operators with Adaptive Hybrid Kernels in Multiscale Sobolev Spaces

Abstract

This paper introduces significant advancements in fractional neural operators (FNOs) through the integration of adaptive hybrid kernels and stochastic multiscale analysis. We address several open problems in the existing literature by establishing four foundational theorems. First, we achieve exact bias-variance separation for Riemann-Liouville operators using fractional Prokhorov metrics, providing a robust framework for handling non-local dependencies in differential equations. Second, we demonstrate L\'evy-regularized Hadamard stability with sharp convergence rates in Besov-Morrey spaces, enhancing FNO stability under heavy-tailed noise processes. Third, we overcome the curse of dimensionality in multivariate settings by achieving tensorized RL-Caputo convergence in RdRd with anisotropic H\"older regularity. Finally, we develop a quantum-inspired fractional gradient descent algorithm that significantly improves convergence rates in practical applications. Our proofs employ advanced techniques such as multiphase homogenization, Malliavin-Skorokhod calculus, and nonlocal divergence theorems, ensuring mathematical rigor and robustness. The practical implications of our work are demonstrated through applications in fusion plasma turbulence, where our methods yield a 70\% improvement over state-of-the-art FNOs. This enhancement is particularly notable in modeling the multifractal {\delta}B{\delta}B fields in ITER tokamak data, where the anisotropic penalty corresponds to enstrophy cascade rates in Hasegawa-Wakatani models, showcasing its versatility and potential for broader application.