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In the present article, we extend the results of the neural network interpolation operators activated by smooth ramp functions proposed by Yu (Acta Math. Sin.(Chin. Ed.) 59:623-638, 2016). We give different results from Yu (Acta Math. Sin.(Chin. Ed.) 59:623-638, 2016) we discuss the high-order approximation result using the smoothness of φ and a related Voronovskaya-type asymptotic expansion for the error of approximation. In addition, we showcase the related fractional estimates result and the fractional Voronovskaya type asymptotic expansion. We investigate the approximation degree for the iterated and complex extensions of the aforementioned operators. Finally, we provide numerical examples and graphs to effectively illustrate and validate our results.

We live in a technological generation surrounded by interconnected sensors that can collect and distribute immense amounts of data on a daily basis. These data would have a better connotation and would have been more practical if sensor-based networks allowed us to capture and monitor the characteristics of physical objects from a highly dynamic environment. At this point, sensor-based networks could substantially enhance their applicability if machines process and interpret vast amounts of data correctly, an essential characteristic of scalable and interoperable wireless sensor network architectures. Through this research project, a) We will identify and evaluate wireless sensor network architectures enabling ap- plications from a highly dynamic environment. Then a data architecture will be proposed to enhance machine-to-machine (M2M) communication and human understanding, considering the issues and challenges of sensor networks. The future proposed data architecture will overcome the existing data frameworks' limitations identified in the literature review. b) The significant contribution of the research is to propose energy-efficient data collection models (the first layer of data architecture) that will reduce data transmissions using prediction models between nodes in sensor networks. The proposed models intend to predict values at the sink node using coefficients built and transmitted by sensor platforms. The goal is to build models that improve the energy of battery-powered sensory devices by reducing data transmissions and recovering values at sink nodes using the same coefficients of models while ensuring data integrity. Furthermore, the models are evaluated using real data sets from real sensor networks with the following metrics; RMSE, MAE, MSE, data reduction percentage, and energy savings.