arXiv subjects

Jorge Henrique de Oliveira Sales

Publications and source records attributed to Jorge Henrique de Oliveira Sales.

18 recordsLinked to original sources

arXiv2025

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

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.

cs.LG↗

arXiv2025

Revolutionizing Fractional Calculus with Neural Networks: Voronovskaya-Damasclin Theory for Next-Generation AI Systems

This work introduces rigorous convergence rates for neural network operators activated by symmetrized and perturbed hyperbolic tangent functions, utilizing novel Voronovskaya-Damasclin asymptotic expansions. We analyze basic, Kantorovich, and quadrature-type operators over infinite domains, extending classical approximation theory to fractional calculus via Caputo derivatives. Key innovations include parameterized activation functions with asymmetry control, symmetrized density operators, and fractional Taylor expansions for error analysis. The main theorem demonstrates that Kantorovich operators achieve $o(n^{-β(N-\varepsilon)})$ convergence rates, while basic operators exhibit $\mathcal{O}(n^{-βN})$ error decay. For deep networks, we prove $\mathcal{O}(L^{-β(N-\varepsilon)})$ approximation bounds. Stability results under parameter perturbations highlight operator robustness. By integrating neural approximation theory with fractional calculus, this work provides foundational mathematical insights and deployable engineering solutions, with potential applications in complex system modeling and signal processing.

cs.LG↗

arXiv2025

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

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.

math.GM↗

arXiv2025

Extension of Symmetrized Neural Network Operators with Fractional and Mixed Activation Functions

We propose a novel extension to symmetrized neural network operators by incorporating fractional and mixed activation functions. This study addresses the limitations of existing models in approximating higher-order smooth functions, particularly in complex and high-dimensional spaces. Our framework introduces a fractional exponent in the activation functions, allowing adaptive non-linear approximations with improved accuracy. We define new density functions based on $q$-deformed and $θ$-parametrized logistic models and derive advanced Jackson-type inequalities that establish uniform convergence rates. Additionally, we provide a rigorous mathematical foundation for the proposed operators, supported by numerical validations demonstrating their efficiency in handling oscillatory and fractional components. The results extend the applicability of neural network approximation theory to broader functional spaces, paving the way for applications in solving partial differential equations and modeling complex systems.

stat.ML↗

arXiv2025

Non-Linear Interactions in Neural Network Operators: New Theorems on Symmetry-Preserving Transformations

This paper advances the study of multivariate function approximation using neural network operators activated by symmetrized and perturbed hyperbolic tangent functions. We propose new non-linear operators that preserve dynamic symmetries within Euclidean spaces, extending current results on Voronovskaya-type asymptotic expansions. The developed operators utilize parameterized deformations to model higher-order interactions in multivariate settings, achieving improved accuracy and robustness. Fundamental theorems demonstrate convergence properties and quantify error bounds, highlighting the operators' ability to approximate functions and derivatives in Sobolev spaces. These results provide a rigorous foundation for theoretical studies and further applications in symmetry-driven modeling and regularization strategies in machine learning and data analysis.

math.GM↗

arXiv2024

Voronovskaya-Type Asymptotic Expansions and Convergence Analysis for Neural Network Operators in Complex Domains

This paper extends the classical theory of Voronovskaya-type asymptotic expansions to generalized neural network operators defined on non-Euclidean and fractal domains. We introduce and analyze smooth operators activated by modified and generalized hyperbolic tangent functions, extending their applicability to manifold and fractal geometries. Key theoretical results include the preservation of density properties, detailed convergence rates, and asymptotic expansions. Additionally, we explore the role of fractional derivatives in defining neural network operators, which capture non-local behavior and are particularly useful for modeling systems with long-range dependencies or fractal-like structures. Our findings contribute to a deeper understanding of neural network operators in complex, structured spaces, offering robust mathematical tools for applications in signal processing on manifolds and solving partial differential equations on fractals.

math.GM↗

arXiv2024

Generalized Lagrangian Coherent Structures in Finsler Manifolds

This paper introduces a novel theoretical framework for identifying Lagrangian Coherent Structures (LCS) in manifolds with non-constant curvature, extending the theory to Finsler manifolds. By leveraging Riemannian and Finsler geometry, we generalize the deformation tensor to account for geodesic stretching in these complex spaces. The main result demonstrates the existence of invariant surfaces acting as LCS, characterized by dominant eigenvalues of the generalized deformation tensor. We discuss potential applications in astrophysics, relativistic fluid dynamics, and planetary science. This work paves the way for exploring LCS in intricate geometrical settings, offering new tools for dynamical system analysis.

math.GM↗

arXiv2024

Semiclassical and Microlocal Analysis of Energy Dissipation and Cascades in Turbulent Flows

This work presents a comprehensive study of the microlocal energy decomposition and propagation of singularities for semiclassically adjusted dissipative pseudodifferential operators. The analysis focuses on the behavior of energy dissipation in turbulent flows modeled by operators $ P_h $ with symbols $ a(x, ξ) \in S^m(\mathbb{R}^n) $, where $ m < 0 $. Using microlocal partitions of unity, we derive an expression for the energy dissipation rate $ \varepsilon_h $ in both spatial and spectral regions, showing its asymptotic equivalence to a two-dimensional integral over phase space. This framework is then applied to the study of the propagation of singularities in solutions to the equation $ P_h u_h = 0 $, where the wavefront set $ \operatorname{WF}_h(u_h) $ evolves along generalized bicharacteristics of the principal symbol $ p_0(x, ξ) $, with energy dissipation controlled by the imaginary part of $ p_0(x, ξ) $. The results combine microlocal analysis, semiclassical techniques, and symbolic calculus to offer insights into the complex dynamics of wave propagation and energy dissipation in turbulent systems, with implications for both theoretical and applied studies in mathematical physics.

math.GM↗

arXiv2024

Quantum-Inspired Stochastic Modeling and Regularity Analysis in Turbulent Flows

This paper introduces a novel mathematical framework for examining the regularity and energy dissipation properties of solutions to the stochastic Navier-Stokes equations. By integrating Sobolev-Besov hybrid spaces, fractional differential operators, and quantum-inspired modeling techniques, we provide a comprehensive analysis that captures the multiscale and chaotic dynamics inherent in turbulent flows. Central to this framework is a Schrödinger-type operator adapted for fluid dynamics, which encapsulates quantum-scale turbulence effects, thereby elucidating the mechanisms of energy redistribution across scales. Additionally, we develop anisotropic stochastic models with direction-dependent viscosity, characterized by a pseudo-differential operator and a covariance matrix governing directional diffusion. These models more accurately reflect real-world turbulence, where viscosity varies with flow orientation, enhancing both theoretical insights and practical simulation capabilities. Our main contributions include new regularity theorems and rigorous a priori estimates for solutions in Sobolev-Besov spaces, alongside proofs of energy dissipation properties in anisotropic contexts. These findings advance the understanding of fluid turbulence by offering a refined approach to studying scale interactions, stochastic effects, and anisotropy in turbulent flows.

math.AP↗

arXiv2024

Dissipation and Regularity in the Smagorinsky Model with Dynamic Boundaries

This article presents an innovative extension of the Smagorinsky model incorporating dynamic boundary conditions and advanced regularity methods. We formulate the modified Navier-Stokes equations with the Smagorinsky term to model dissipation in turbulence and prove theorems concerning the existence, uniqueness, and asymptotic behavior of solutions. The first theorem establishes the existence and uniqueness of solutions in higher Sobolev spaces, considering the effect of the nonlinear Smagorinsky term and dynamic boundary conditions. The proof employs the Galerkin method and energy estimates, culminating in the application of Grönwall's theorem. The second theorem investigates the asymptotic behavior of solutions, focusing on anomalous dissipation in high-turbulence regimes. We demonstrate that the dissipated energy does not decrease with vanishing viscosity, indicating the occurrence of anomalous dissipation. The third theorem explores advanced regularity in higher Sobolev spaces, allowing for more rigorous control of nonlinear terms and ensuring improved stability conditions. The proof utilizes the energy method combined with Sobolev estimates and Grönwall's inequality. These mathematical results are fundamental for the analysis of dissipation in turbulent flows and can inspire new approaches in numerical simulations of fluids.

math.AP↗

arXiv2024

Adaptive Dissipation in the Smagorinsky Model for Turbulence in Boundary-Driven Flows

This paper enhances the classic Smagorinsky model by introducing an innovative, adaptive dissipation term that adjusts dynamically with distance from boundary regions. This modification addresses a known limitation of the standard model over dissipation near boundaries thereby improving accuracy in turbulent flow simulations in confined or wall-adjacent areas. We present a rigorous theoretical framework for this adaptive model, including two foundational theorems. The first theorem guarantees existence and uniqueness of solutions, ensuring that the model is mathematically well-posed within the adaptive context. The second theorem provides a precise bound on the energy dissipation rate, demonstrating that dissipation remains controlled and realistic even as boundary effects vary spatially. By allowing the dissipation coefficient to decrease near boundary layers, this approach preserves the finer turbulent structures without excessive smoothing, yielding a more physically accurate representation of the flow. Future work will focus on implementing this adaptive model in computational simulations to empirically verify the theoretical predictions and assess performance in scenarios with complex boundary geometries.

physics.flu-dyn↗

arXiv2024

Enhanced Diffusion in Anisotropic and Non-uniform Geometries

Enhanced diffusion, which describes the accelerated spread of passive scalars due to the interaction between advection and molecular diffusion, has been extensively studied in simplified geometries, such as uniform shear and radial flows. However, many real-world applications occur in complex, anisotropic domains where standard assumptions do not hold. This paper extends the theory of enhanced diffusion to anisotropic and non-uniform geometries, where scaling in the $x$- and $y$-directions follows distinct, power-law relationships, and the velocity field exhibits spatially varying regularity. We define a generalized framework for anisotropic diffusion enhancement and rigorously derive new scaling laws for diffusion rates in these settings. Specifically, we show that in a domain with anisotropic scaling functions $ f(x) \sim |x|^p $ and $ g(y) \sim |y|^q $, the enhanced diffusion rate $ r(κ) $ for a passive scalar satisfies $ r(κ) = C κ^{\frac{pq}{p+q+2}} $, where $ C $ depends on the regularity and scaling properties of the domain but remains independent of the diffusivity $ κ$. This theoretical result is validated through detailed proofs leveraging stochastic process techniques and variance estimation. Our findings offer a new perspective on the role of domain geometry and anisotropy in diffusion processes, with potential applications in environmental science, engineering, and beyond. This framework lays the groundwork for future studies in both theoretical and applied settings, enabling a deeper understanding of transport phenomena in complex domains.

math.AP↗

arXiv2024

Stochastic Trajectories and Spectral Boundary Conditions for Enhanced Diffusion in Immersed Boundary Problems

This work presents a comprehensive framework for enhanced diffusion modeling in fluid-structure interactions by combining the Immersed Boundary Method (IBM) with stochastic trajectories and high-order spectral boundary conditions. Using semi-Lagrangian schemes, this approach captures complex diffusion dynamics at moving interfaces, integrating probabilistic methods that reflect multi-scale fluctuations. In addition to a rigorous mathematical foundation that includes stability proofs, this model exhibits reduced numerical diffusion errors and improved stability in long-term simulations. Comparative studies highlight its effectiveness in multi-scale scenarios that require precision in interface dynamics. Focusing on various shear and circular flows, including those with Hölder and Lipschitz regularities and critical points, we establish sharp bounds on effective diffusion rates using specific initial data examples. This dual exploration in enhanced diffusion highlights how flow regularity and critical points influence dissipation. These findings advance both the theoretical understanding and practical applications of enhanced diffusion in fluid dynamics, offering new insights into diffusion rate optimization through interface dynamics and flow structure regularities. Future research can further refine the IBM framework by exploring alternative probabilistic methods to improve interface accuracy, opening up the potential for improved modeling in applications that require precise control over mixing rates and dissipation processes.

math.AP↗

arXiv2024

Hybrid Adaptive Dual Reciprocity Method for Efficient Solution of Large-Scale Non-Linear Boundary Conditions

This article proposes a hybrid adaptive numerical method based on the Dual Reciprocity Method (DRM) to solve problems with non-linear boundary conditions and large-scale problems, named Hybrid Adaptive Dual Reciprocity Method (H-DRM). The method uses a combination of DRM to handle non-homogeneous terms, iterative techniques to deal with non-linear boundary conditions, and an adaptive multiscale approach for large-scale problems. Additionally, the H-DRM incorporates local finite elements in critical regions of the domain. This method aims to improve computational efficiency and accuracy for problems involving complex geometry and non-linearities at the boundary, offering a robust solution for physical and engineering problems. Demonstrations and computational results are presented, validating the effectiveness of the method compared to other known methods through an iterative process of 7 million iterations.

math.NA↗

arXiv2024

Vortex Formation and Dissipation in Chaotic Flows: A Hypercomplex Approach

This article presents a comprehensive analysis of the formation and dissipation of vortices within chaotic fluid flows, leveraging the framework of Sobolev and Besov spaces on Riemannian manifolds. Building upon the Navier-Stokes equations, we introduce a hypercomplex bifurcation approach to characterize the regularity and critical thresholds at which vortices emerge and dissipate in chaotic settings. We explore the role of differential geometry and bifurcation theory in vortex dynamics, providing a rigorous mathematical foundation for understanding these phenomena. Our approach addresses spectral decomposition, asymptotic stability, and dissipation thresholds, offering critical insights into the mechanisms of vortex formation and dissipation. Additionally, we introduce two new theorems that further elucidate the role of geometric stability and bifurcations in vortex dynamics. The first theorem demonstrates the geometric stability of vortices on manifolds with positive Ricci curvature, while the second theorem analyzes the bifurcation points that lead to the formation or dissipation of vortex structures. This research contributes to the existing literature by providing a more complete mathematical picture of the underlying mechanisms of vortex dynamics in turbulent fluid flows.

physics.flu-dyn↗

arXiv2024

High-Order Dynamic Integration Method (HODIM) for Modeling Turbulent Fluid Dynamics

This research explores the development and application of the High-Order Dynamic Integration Method for solving integro-differential equations, with a specific focus on turbulent fluid dynamics. Traditional numerical methods, such as the Finite Difference Method and the Finite Volume Method, have been widely employed in fluid dynamics but struggle to accurately capture the complexities of turbulence, particularly in high Reynolds number regimes. These methods often require significant computational resources and are prone to errors in nonlinear dynamic systems. The High-Order Dynamic Integration Method addresses these challenges by integrating higher-order interpolation techniques with dynamic adaptation strategies, significantly enhancing accuracy and computational efficiency. Through rigorous numerical analysis, this method demonstrates superior performance over the Finite Difference Method and the Finite Volume Method in handling the nonlinear behaviors characteristic of turbulent flows. Furthermore, the High-Order Dynamic Integration Method achieves this without a substantial increase in computational cost, making it a highly efficient tool for simulations in computational fluid dynamics. The research validates the capabilities of the High-Order Dynamic Integration Method through a series of benchmark tests and case studies. Results indicate a marked improvement in both accuracy and stability, particularly in simulations of high-Reynolds-number flows, where traditional methods often falter. This innovative approach offers a robust and efficient alternative for solving complex fluid dynamics problems, contributing to advances in the field of numerical methods and computational fluid dynamics.

physics.flu-dyn↗

arXiv2024

Quantum-Inspired Stochastic Modeling and Regularity in Turbulent Fluid Dynamics

This paper presents an innovative framework for analyzing the regularity of solutions to the stochastic Navier-Stokes equations by integrating Sobolev-Besov hybrid spaces with fractional operators and quantum-inspired dynamics. We propose new regularity theorems that address the multiscale and chaotic nature of fluid flows, offering novel insights into energy dissipation mechanisms. The introduction of a Schrödinger-like operator into the fluid dynamics model captures quantum-scale turbulence effects, enhancing our understanding of energy redistribution across different scales. The results also include the development of anisotropic stochastic models that account for direction-dependent viscosity, improving the representation of real-world turbulent flows. These advances in stochastic modeling and regularity analysis provide a comprehensive toolset for tackling complex fluid dynamics problems. The findings are applicable to fields such as engineering, quantum turbulence, and environmental sciences. Future directions include the numerical implementation of the proposed models and the use of machine learning techniques to optimize parameters for enhanced simulation accuracy.

physics.flu-dyn↗

arXiv2024

Hypercomplex Dynamics and Turbulent Flows in Sobolev and Besov Functional Spaces

This paper presents a rigorous study of advanced functional spaces, with a focus on Sobolev and Besov spaces, to investigate key aspects of fluid dynamics, including the regularity of solutions to the Navier-Stokes equations, hypercomplex bifurcations, and turbulence. We offer a comprehensive analysis of Sobolev embedding theorems in fractional spaces and apply bifurcation theory within quaternionic dynamical systems to better understand the complex behaviors in fluid systems. Additionally, the research delves into energy dissipation mechanisms in turbulent flows through the framework of Besov spaces. Key mathematical tools, such as interpolation theory, Littlewood-Paley decomposition, and energy cascade models, are integrated to develop a robust theoretical approach to these problems. By addressing challenges related to the existence and smoothness of solutions, this work contributes to the ongoing exploration of the open Navier-Stokes problem, providing new insights into the intricate relationship between fluid dynamics and functional spaces.

math.AP↗