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

Romulo Damaselin Chaves dos Santos, Jorge Henrique de Oliveira Sales

TL;DR

This work addresses core limitations of fractional neural operators (FNOs) by introducing spatially adaptive, hybrid RL–Caputo derivatives and Lévy-regularized Hadamard kernels to achieve exact bias–variance separation and robust stability under heavy-tailed noise. It develops a tensorized, anisotropically regularized multiscale framework that overcomes the curse of dimensionality and provides sharp convergence in Hölder and Besov spaces, underpinned by Malliavin–Skorokhod calculus and nonlocal divergence theorems. A quantum-inspired fractional gradient descent (QFGD) algorithm is proposed, yielding dimension-independent convergence improvements and accelerated optimization in high-dimensional turbulent settings. Empirical validation on ITER tokamak data shows a 70% reduction in modeling error for multifractal δB fields and spectral alignment with enstrophy-cascade dynamics, demonstrating practical impact in fusion-plasma modeling and broader scientific computing contexts.

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 Lévy-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 Hölder 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 δBδB 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.

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

TL;DR

This work addresses core limitations of fractional neural operators (FNOs) by introducing spatially adaptive, hybrid RL–Caputo derivatives and Lévy-regularized Hadamard kernels to achieve exact bias–variance separation and robust stability under heavy-tailed noise. It develops a tensorized, anisotropically regularized multiscale framework that overcomes the curse of dimensionality and provides sharp convergence in Hölder and Besov spaces, underpinned by Malliavin–Skorokhod calculus and nonlocal divergence theorems. A quantum-inspired fractional gradient descent (QFGD) algorithm is proposed, yielding dimension-independent convergence improvements and accelerated optimization in high-dimensional turbulent settings. Empirical validation on ITER tokamak data shows a 70% reduction in modeling error for multifractal δB fields and spectral alignment with enstrophy-cascade dynamics, demonstrating practical impact in fusion-plasma modeling and broader scientific computing contexts.

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 Lévy-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 Hölder 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 δBδB 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.

Paper Structure

This paper contains 35 sections, 6 theorems, 55 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Mathematical Background
  3. Fractional Calculus
  4. Functional Analysis
  5. Stochastic Processes
  6. Multiphase Homogenization
  7. Malliavin-Skorokhod Calculus
  8. Nonlocal Divergence Theorems
  9. ITER Tokamak Data & Enstrophy Cascade
  10. Mathematical Foundations
  11. Adaptive Hybrid Derivatives in Negative Hölder Spaces
  12. Lévy-Regularized Hadamard Kernels
  13. Functional Analytic Framework
  14. Main Theorem
  15. Adaptive Hybrid Kernel Convergence
  16. ...and 20 more sections

Key Result

Theorem 4.1

Let $f \in \mathcal{F}(\Omega)$, $\alpha \in C^{1,\delta}(\Omega)$ with $\delta > 1/2$, and $0 < \alpha_0 \leq \alpha(x) \leq \alpha_1 < 1$. Then for $C_n^{Adapt}$ using $\Phi_H^L$:

Figures (2)

  • Figure 1: Comparison of error reduction between adaptive hybrid kernels and traditional FNOs in modeling $\delta B$ fields.
  • Figure 2: Comparison of error reduction using QFGD, traditional gradient descent, and FNOs. QFGD achieves faster convergence due to the fractional update mechanism.

Theorems & Definitions (12)

  • Theorem 4.1: Adaptive Hybrid Kernel Convergence
  • proof
  • Theorem 4.2: Multiscale Convergence
  • proof
  • Theorem 4.3: Fractional Embedding in Besov Spaces
  • proof
  • Theorem 4.4: Convergence in Hölder Spaces
  • proof
  • Theorem 4.5: Fractional Sobolev Embedding
  • proof
  • ...and 2 more