通过分数阶与混合激活函数扩展神经网络算子，提升高维光滑函数逼近能力。
Extension of Symmetrized Neural Network Operators with Fractional and Mixed Activation Functions
- 引入分数阶激活函数，实现自适应非线性逼近
- 建立新型密度函数与杰克逊型不等式，保证一致收敛速率
- 适合需高精度逼近的偏微分方程与复杂系统建模场景
我们提出一种新方法，通过引入分数阶和混合激活函数，扩展对称神经网络算子。该研究针对现有模型在复杂高维空间中逼近高阶光滑函数的局限性，提出在激活函数中引入分数指数，实现更精确的自适应非线性逼近。基于 $q$-变形和 $θ$-参数化逻辑模型构建新的密度函数，并推导出高级杰克逊型不等式，确立统一收敛速率。同时提供严格的数学基础，数值验证表明其在处理振荡与分数成分时具有高效性。研究成果将神经网络逼近理论的应用范围拓展至更广泛的函数空间，为求解偏微分方程和建模复杂系统开辟新路径。
原文摘要 · Abstract (English)
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.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。