用神经网络提升分数阶微积分精度，理论证明收敛速度更快。
Revolutionizing Fractional Calculus with Neural Networks: Voronovskaya-Damasclin Theory for Next-Generation AI Systems
- 设计带不对称控制的激活函数，改进神经算子在无限域上的逼近能力。
- 证明Kantorovich算子收敛率达$o(n^{-β(N-ε)})$，深度网络达$O(L^{-β(N-ε)})$。
- 为复杂系统建模与信号处理提供可部署的数学基础，适合算法研究者。
本文提出基于对称化及扰动双曲正切激活函数的神经网络算子收敛率理论，引入新颖的Voronovskaya-Damasclin渐近展开。研究覆盖基本型、Kantorovich型与求积型算子在无限域上的表现，通过Caputo导数将经典逼近论拓展至分数阶微积分领域。关键创新包括具参数化的不对称激活函数、对称密度算子及用于误差分析的分数阶泰勒展开。主定理表明，Kantorovich算子收敛率为$o(n^{-β(N-ε)})$，基本算子为$/mathcal{O}(n^{-βN})$；对深度网络，证明了$/mathcal{O}(L^{-β(N-ε)})$的逼近界。参数扰动下的稳定性分析揭示算子鲁棒性。本工作融合神经逼近理论与分数阶微积分，提供基础数学洞察与可部署工程方案，适用于复杂系统建模与信号处理。
原文摘要 · Abstract (English)
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.
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