Approximation Analysis of Symmetry-Enhanced Neural Network Operators by Machine Learning Metrics

This paper investigates the approximation properties of both classical and fractional neural network (NN) operators and their symmetrized counterparts (SNN), which are activated by an activation function, half-hyperbolic tangent, acting on Banach space-valued functions, f :X → R, where X is a Banach space with norm ||·||. This work is motivated by the need to better understand the mathematical behavior of neural network (NN)-type operators within the framework of approximation theory and functional analysis. Within this setting, we establish pointwise and uniform convergence results together with quantitative fractional approximation estimates expressed in terms of suitable smoothness measures. A central contribution of the paper is a rigorous comparison between classical neural network (NN) operators and their symmetrized counterparts (SNN). The analysis shows that the symmetry structure leads to improved approximation properties and enhanced stability under appropriate parameter regimes. One of the most innovative aspects of this paper is the evaluation of approximation operators using regression-based machine learning metrics, including RMSE, MAE, Maximum error, and R². These metrics provide a statistical interpretation of approximation quality, complementing classical norm-based error bounds. This approach is important because it connects approximation theory with data science methodology and makes operator performance interpretable for machine learning applications. To complement the theoretical analysis, we conducted systematic numerical experiments using the approximation of the proposed operators to the test functions, a log–log graph, and an error-decay graph implemented in Python 3.13, including graphical illustrations and quantitative comparisons of NN and SNN operators employing machine learning metrics. The computational results support the theoretical findings and demonstrate that SNN operators can achieve improved approximation accuracy and more stable behavior in practice. To promote transparency and reproducibility, all implementation details and Python codes, algorithms were made publicly available via GitHub. By combining rigorous analysis with computational experiments, this work contributes to strengthening the connection between approximation theory, neural network models, and numerical computation, providing both theoretical insights and practical tools for researchers working at the interface of mathematics and machine learning.