Extension of Symmetrized Neural Network Operators with Fractional and Mixed Activation Functions
This work addresses limitations in neural network approximation theory for researchers in applied mathematics and machine learning, though it appears incremental as it builds on existing symmetrized operators.
The authors tackled the problem of approximating higher-order smooth functions in complex, high-dimensional spaces by extending symmetrized neural network operators with fractional and mixed activation functions, resulting in improved accuracy and uniform convergence rates as demonstrated through numerical validations.
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.