The Sigmoid Neural Network Activation Function and its Connections to Airy’s and the Nield-Kuznetsov Functions

Abstract

Analysis and solution of Airy’s inhomogeneous equation, when its forcing function is the sigmoid neural network activation function, are provided in this work. Relationship between the Nield-Kuznetsov, the Scorer, the sigmoid, the polylogarithm and Airy’s functions are established. Solutions to initial and boundary value problems, when the sigmoid function is involved, are obtained. Computations were carried out using Wolfram Alpha.

Connecting Einstein Functions to the Nield-Kuznetsov and Airy’s Functions

“…Noteworthy in the study of Einstein functions is their connection to polylogarithmic functions, [7,8], which bridge a gap in our mathematical knowledge between Airy's inhomoheneous ordinary differential equation (ODE) with homogeneities due to special functions, such as the sigmoid logistic function. This connection was recently studied and established by Roach and Hamdan, [9], and Hamdan and Roach, [10], whose work underscored the importance of connections between Airy's functions, special functions and the Nield-Kuznetsov integral functions. Their work has inevitably lead to the current work where a connection is being sought between the Einstein functions, the Nield-Kuznetsov functions and the classic Airy's functions, [11,12].…”

“…General solution to (10) is of the form 𝑦 = 𝑐 1 𝐴𝑖(𝑥) + 𝑐 2 𝐵𝑖(𝑥) + 𝑦 𝑝 (11) where 𝑦 𝑝 is given by the following equivalent forms:…”

“…This approach is robust and versatile, and both forms, ( 6) and (7), have recently been used to obtain particular solutions to (1) when 𝑓(𝑥) = 𝐴 𝑖 (𝑥), 𝐵 𝑖 (𝑥), or 𝑁 𝑖 (𝑥), [16], and when 𝑓(𝑥) = 𝑆(𝑥) is the sigmoid logistic function, [10].…”

“…It is worth noting that the work of Hamdan and Roach, [10], helped to establish a mathematically significant connection between Airy's functions, the sigmoid functions and the dilogarithm function, and between the sigmoid function and the Nield-Kuznetsov functions, 𝑁𝑖(𝑥) and 𝐾𝑖(𝑥). These forms of forcing functions gave rise to interesting integrals involving products of these special functions, and introduced new functions, such as the dilogarithm function, as building blocks of solutions to Airy's ODE.…”