Soboleva modified hyperbolic tangent

The Soboleva modified hyperbolic tangent, also known as (parametric) Soboleva modified hyperbolic tangent activation function ([P]SMHTAF), is a special S-shaped function based on the hyperbolic tangent, given by

History
This function was originally proposed as "modified hyperbolic tangent" by Ukrainian scientist Elena V. Soboleva (Елена В. Соболева) as a utility function for multi-objective optimization and choice modelling in decision-making.

Practical usage
The function has since been introduced into neural network theory and practice.

It was also used in economics for modelling consumption and investment, to approximate current-voltage characteristics of field-effect transistors and light-emitting diodes, to design antenna feeders, and analyze plasma temperatures and densities in the divertor region of fusion reactors.

Sensitivity to parameters
Derivative of the function is defined by the formula:

$$\operatorname{smht}'(x) \doteq \frac {ae^{ax} + be^{-bx}} {e^{cx} + e^{-dx}} - \operatorname{smht}(x)\frac {ce^{cx} - de^{-dx}} {e^{cx} + e^{-dx}}$$

The following conditions are keeping the function limited on y-axes: a ≤ c, b ≤ d.

A family of recurrence-generated parametric Soboleva modified hyperbolic tangent activation functions (NPSMHTAF, FPSMHTAF) was studied with parameters a = c and b = d. It is worth noting that in this case, the function is not sensitive to flipping the left and right-sides parameters:

The function is sensitive to ratio of the denominator coefficients and often is used without coefficients in the numerator:

With parameters a = b = c = d = 1 the modified hyperbolic tangent function reduces to the conventional tanh(x) function, whereas for a = b = 1 and c = d = 0, the term becomes equal to sinh(x).