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Stochastic Fractional Neural Operators: A Symmetrized Approach to Modeling Turbulence in Complex Fluid Dynamics

Rômulo Damasclin Chaves dos Santos, Jorge Henrique de Oliveira Sales

TL;DR

This work develops stochastic, symmetrized neural operators that integrate Caputo fractional memory and Itô-type noise to approximate time-evolving functions with nonlocal memory and uncertainty. By constructing a symmetrized kernel and a stochastic Kantorovich-type operator K_n^W, the authors establish Voronovskaya-type expansions and mean-square convergence results, with explicit dependence on the memory parameter $α$ and noise level $σ$. They apply the framework to fractional Navier–Stokes equations with stochastic forcing, deriving deterministic bias results, variance characterizations, energy-dissipation convergence, and $L^2$-convergence guarantees. The approach blends neural approximation with fractional calculus and stochastic analysis, offering a principled tool for modeling turbulence and other multiscale systems where memory and randomness are fundamental.

Abstract

In this work, we introduce a new class of neural network operators designed to handle problems where memory effects and randomness play a central role. In this work, we introduce a new class of neural network operators designed to handle problems where memory effects and randomness play a central role. These operators merge symmetrized activation functions, Caputo-type fractional derivatives, and stochastic perturbations introduced via Itô type noise. The result is a powerful framework capable of approximating functions that evolve over time with both long-term memory and uncertain dynamics. We develop the mathematical foundations of these operators, proving three key theorems of Voronovskaya type. These results describe the asymptotic behavior of the operators, their convergence in the mean-square sense, and their consistency under fractional regularity assumptions. All estimates explicitly account for the influence of the memory parameter $α$ and the noise level $σ$. As a practical application, we apply the proposed theory to the fractional Navier-Stokes equations with stochastic forcing, a model often used to describe turbulence in fluid flows with memory. Our approach provides theoretical guarantees for the approximation quality and suggests that these neural operators can serve as effective tools in the analysis and simulation of complex systems. By blending ideas from neural networks, fractional calculus, and stochastic analysis, this research opens new perspectives for modeling turbulent phenomena and other multiscale processes where memory and randomness are fundamental. The results lay the groundwork for hybrid learning-based methods with strong analytical backing.

Stochastic Fractional Neural Operators: A Symmetrized Approach to Modeling Turbulence in Complex Fluid Dynamics

TL;DR

This work develops stochastic, symmetrized neural operators that integrate Caputo fractional memory and Itô-type noise to approximate time-evolving functions with nonlocal memory and uncertainty. By constructing a symmetrized kernel and a stochastic Kantorovich-type operator K_n^W, the authors establish Voronovskaya-type expansions and mean-square convergence results, with explicit dependence on the memory parameter and noise level . They apply the framework to fractional Navier–Stokes equations with stochastic forcing, deriving deterministic bias results, variance characterizations, energy-dissipation convergence, and -convergence guarantees. The approach blends neural approximation with fractional calculus and stochastic analysis, offering a principled tool for modeling turbulence and other multiscale systems where memory and randomness are fundamental.

Abstract

In this work, we introduce a new class of neural network operators designed to handle problems where memory effects and randomness play a central role. In this work, we introduce a new class of neural network operators designed to handle problems where memory effects and randomness play a central role. These operators merge symmetrized activation functions, Caputo-type fractional derivatives, and stochastic perturbations introduced via Itô type noise. The result is a powerful framework capable of approximating functions that evolve over time with both long-term memory and uncertain dynamics. We develop the mathematical foundations of these operators, proving three key theorems of Voronovskaya type. These results describe the asymptotic behavior of the operators, their convergence in the mean-square sense, and their consistency under fractional regularity assumptions. All estimates explicitly account for the influence of the memory parameter and the noise level . As a practical application, we apply the proposed theory to the fractional Navier-Stokes equations with stochastic forcing, a model often used to describe turbulence in fluid flows with memory. Our approach provides theoretical guarantees for the approximation quality and suggests that these neural operators can serve as effective tools in the analysis and simulation of complex systems. By blending ideas from neural networks, fractional calculus, and stochastic analysis, this research opens new perspectives for modeling turbulent phenomena and other multiscale processes where memory and randomness are fundamental. The results lay the groundwork for hybrid learning-based methods with strong analytical backing.

Paper Structure

This paper contains 22 sections, 3 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Symmetrized Activation Functions and Stochastic Perturbations
  3. Analytical Properties of the Symmetrized Kernel and Operator
  4. Symmetry and Positivity
  5. Regularity and Decay
  6. Approximate Partition of Unity
  7. Linearity and Stochasticity
  8. Noise-Induced Variability
  9. Caputo Derivatives and Fractional Smoothness
  10. Main Theorems
  11. Application: Turbulence in Fractional Navier-Stokes
  12. Approximation Using the Stochastic Kantorovich Operator
  13. Deterministic Convergence (Bias Analysis)
  14. Stochastic Fluctuation (Variance Analysis)
  15. Mean Squared Error (MSE) Analysis
  16. ...and 7 more sections

Key Result

Theorem 4.1

Let $\mathrm{f} \in C^m(\mathbb{R}^N) \cap W^{\alpha,\infty}(\mathbb{R}^N)$ for some $m \geq 2$ and $0 < \alpha < 1$. Then: where $M_{\beta,n}$ are moments of the kernel $\mathcal{Z}$.

Theorems & Definitions (3)

  • Theorem 4.1: Fractional-Stochastic Voronovskaya Expansion
  • Theorem 4.2: Mean-Square Convergence
  • Theorem 4.3: Fractional Stochastic Kantorovich Consistency