Soboleva modified hyperbolic tangent

The Soboleva modified hyperbolic tangent, also known as (parametric) Soboleva modified hyperbolic tangent activation function ([P]SMHTAF),[nb 1] is a special S-shaped function based on the hyperbolic tangent, given by

${\displaystyle \operatorname {mtanh} x={\frac {e^{ax}-e^{-bx}}{e^{cx}+e^{-dx}}}.}$

This function was originally proposed as "modified hyperbolic tangent"[nb 1] by Elena V. Soboleva (Елена В. Соболева) as a utility function for multi-objective optimization and choice modelling in decision-making.[1][2][3] It has since been introduced into neural network theory and practice.[4][5]

The function was also used to approximate current-voltage characteristics of field-effect transistors and light-emitting diodes,[6] to design antenna feeders,[7][4] and analyze plasma temperatures and densities in the divertor region of fusion reactors.[8]

A family of recurrence-generated parametric Soboleva modified hyperbolic tangent activation functions (NPSMHTAF, FPSMHTAF) was studied with parameters a = c and b = d.[4]

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).