  Selecting the most suitable activation function is a critical factor in the
effectiveness of deep learning models, as it influences their learning
capacity, stability, and computational efficiency. In recent years, the
Gaussian Error Linear Unit (GELU) activation function has emerged as a dominant
method, surpassing traditional functions such as the Rectified Linear Unit
(ReLU) in various applications. This study presents a rigorous mathematical
investigation of the GELU activation function, exploring its differentiability,
boundedness, stationarity, and smoothness properties in detail. Additionally,
we conduct an extensive experimental comparison of the GELU function against a
broad range of alternative activation functions, utilizing a residual
convolutional network trained on the CIFAR-10, CIFAR-100, and STL-10 datasets
as the empirical testbed. Our results demonstrate the superior performance of
GELU compared to other activation functions, establishing its suitability for a
wide range of deep learning applications. This comprehensive study contributes
to a more profound understanding of the underlying mathematical properties of
GELU and provides valuable insights for practitioners aiming to select
activation functions that optimally align with their specific objectives and
constraints in deep learning.
