Neural network activation functions

[Beliavsky]

I think neural networks may be too broad a topic for stdlib (there are a number of Fortran projects in this area), but activation functions and their derivatives could be considered.

[jalvesz]

That's a good idea, maybe something like this could be a starting point:

Click me: stdlib_math_activations.fypp

#:include "common.fypp"
module stdlib_math_activations
    use stdlib_kinds, only: int8, int16, int32, int64, sp, dp, xdp, qp
    implicit none
    private

    interface gaussian
        #:for k1, t1 in REAL_KINDS_TYPES
        module procedure :: gaussian_${k1}$
        #:endfor
    end interface
    public :: gaussian

    interface gaussian_grad
        #:for k1, t1 in REAL_KINDS_TYPES
        module procedure :: gaussian_grad_${k1}$
        #:endfor
    end interface
    public :: gaussian_grad

    interface elu
        #:for k1, t1 in REAL_KINDS_TYPES
        module procedure :: elu_${k1}$
        #:endfor
    end interface
    public :: elu

    interface elu_grad
        #:for k1, t1 in REAL_KINDS_TYPES
        module procedure :: elu_grad_${k1}$
        #:endfor
    end interface
    public :: elu_grad

    interface relu
        #:for k1, t1 in REAL_KINDS_TYPES
        module procedure :: relu_${k1}$
        #:endfor
    end interface
    public :: relu

    interface relu_grad
        #:for k1, t1 in REAL_KINDS_TYPES
        module procedure :: relu_grad_${k1}$
        #:endfor
    end interface
    public :: relu_grad

    interface gelu
        #:for k1, t1 in REAL_KINDS_TYPES
        module procedure :: gelu_${k1}$
        #:endfor
    end interface
    public :: gelu

    interface gelu_grad
        #:for k1, t1 in REAL_KINDS_TYPES
        module procedure :: gelu_grad_${k1}$
        #:endfor
    end interface
    public :: gelu_grad

    interface gelu_approx
        #:for k1, t1 in REAL_KINDS_TYPES
        module procedure :: gelu_approx_${k1}$
        #:endfor
    end interface
    public :: gelu_approx

    interface gelu_approx_grad
        #:for k1, t1 in REAL_KINDS_TYPES
        module procedure :: gelu_approx_grad_${k1}$
        #:endfor
    end interface
    public :: gelu_approx_grad

    interface sigmoid
        #:for k1, t1 in REAL_KINDS_TYPES
        module procedure :: sigmoid_${k1}$
        #:endfor
    end interface
    public :: sigmoid

    interface sigmoid_grad
        #:for k1, t1 in REAL_KINDS_TYPES
        module procedure :: sigmoid_grad_${k1}$
        #:endfor
    end interface
    public :: sigmoid_grad

    interface step
        #:for k1, t1 in REAL_KINDS_TYPES
        module procedure :: step_${k1}$
        #:endfor
    end interface
    public :: step

    interface step_grad
        #:for k1, t1 in REAL_KINDS_TYPES
        module procedure :: step_grad_${k1}$
        #:endfor
    end interface
    public :: step_grad

    interface Softmax
        #:for k1, t1 in REAL_KINDS_TYPES
        module procedure :: softmax_${k1}$
        #:endfor
    end interface
    public :: softmax

    interface Softmax_grad
        #:for k1, t1 in REAL_KINDS_TYPES
        module procedure :: Softmax_grad_${k1}$
        #:endfor
    end interface
    public :: Softmax_grad

    interface Softplus
        #:for k1, t1 in REAL_KINDS_TYPES
        module procedure :: Softplus_${k1}$
        #:endfor
    end interface
    public :: Softplus

    interface Softplus_grad
        #:for k1, t1 in REAL_KINDS_TYPES
        module procedure :: Softplus_grad_${k1}$
        #:endfor
    end interface
    public :: Softplus_grad
    
    #:for k1, t1 in REAL_KINDS_TYPES
    ${t1}$, parameter :: isqrt2_${k1}$ = 1_${k1}$ / sqrt(2._${k1}$)
    #:endfor

contains

!==================================================
! Gaussian
!==================================================
#:for k1, t1 in REAL_KINDS_TYPES
elemental ${t1}$ function gaussian_${k1}$( x ) result( y )
    ${t1}$, intent(in) :: x
    !==================================================
    y = exp(-x**2)
end function

elemental ${t1}$ function gaussian_grad_${k1}$( x ) result( y )
    ${t1}$, intent(in) :: x
    !==================================================
    y = -2_${k1}$ * x * exp(-x**2)
end function

#:endfor

!==================================================
! Exponential Linear Unit
!==================================================
#:for k1, t1 in REAL_KINDS_TYPES
elemental ${t1}$ function elu_${k1}$( x , a ) result ( y )
    ${t1}$, intent(in) :: x
    ${t1}$, intent(in) :: a
    !==================================================
    if(x >= 0_${k1}$)then
        y = x
    else
        y = a * (exp(x) - 1_${k1}$)
    end if
end function

elemental ${t1}$ function elu_grad_${k1}$( x , a ) result ( y )
    ${t1}$, intent(in) :: x
    ${t1}$, intent(in) :: a
    !==================================================
    if(x >= 0_${k1}$)then
        y = 1_${k1}$ 
    else
        y = a * exp(x)
    end if
end function

#:endfor

!==================================================
! Rectified Linear Unit
!==================================================
#:for k1, t1 in REAL_KINDS_TYPES
elemental ${t1}$ function relu_${k1}$( x ) result( y )
    ${t1}$, intent(in) :: x
    !==================================================
    y = max(0._${k1}$, x)
end function

elemental ${t1}$ function relu_grad_${k1}$( x ) result( y )
    ${t1}$, intent(in) :: x
    !==================================================
    if(x > 0_${k1}$)then
        y = 1_${k1}$ 
    else
        y = 0_${k1}$
    end if
end function

#:endfor

!==================================================
! GELU: Gaussian Error Linear Units function
!==================================================
#:for k1, t1 in REAL_KINDS_TYPES
elemental ${t1}$ function gelu_${k1}$( x ) result( y )
    ${t1}$, intent(in) :: x
    !==================================================
    y = 0.5_${k1}$ * x * (1 + erf(x * isqrt2_${k1}$))
end function

elemental ${t1}$ function gelu_grad_${k1}$( x ) result( y )
    ${t1}$, intent(in) :: x
    !==================================================
    y = 0.5_${k1}$ * (1 + erf(x * isqrt2_${k1}$) )
    y = y + x * isqrt2_${k1}$ * exp( - 0.5_${k1}$ * x**2 )
end function

#:endfor

#:for k1, t1 in REAL_KINDS_TYPES
elemental ${t1}$ function gelu_approx_${k1}$( x ) result( y )
    ${t1}$, intent(in) :: x
    !==================================================
    y = 0.5_${k1}$ * x * (1 + erf(x * isqrt2_${k1}$))
end function

elemental ${t1}$ function gelu_approx_grad_${k1}$( x ) result( y )
    ${t1}$, intent(in) :: x
    !==================================================
    y = 0.5_${k1}$ * (1 + erf(x * isqrt2_${k1}$) )
    y = y + x * isqrt2_${k1}$ * exp( - 0.5_${k1}$ * x**2 )
end function

#:endfor

!==================================================
! Sigmoid
!==================================================
#:for k1, t1 in REAL_KINDS_TYPES
elemental ${t1}$ function sigmoid_${k1}$( x ) result( y )
    ${t1}$, intent(in) :: x
    !==================================================
    y = 1_${k1}$ / (1_${k1}$ + exp(-x))
end function

elemental ${t1}$ function sigmoid_grad_${k1}$( x ) result( y )
    ${t1}$, intent(in) :: x
    !==================================================
    y = exp(x) / (1_${k1}$ + exp(x))**2
end function

#:endfor

!==================================================
! Step
!==================================================
#:for k1, t1 in REAL_KINDS_TYPES
elemental ${t1}$ function Step_${k1}$( x ) result( y )
    ${t1}$, intent(in) :: x
    !==================================================
    if(x > 0_${k1}$)then
        y = 1_${k1}$ 
    else
        y = 0_${k1}$
    end if
end function

elemental ${t1}$ function Step_grad_${k1}$( x ) result( y )
    ${t1}$, intent(in) :: x
    !==================================================
    y = 0_${k1}$
end function

#:endfor

!==================================================
! tanh
!==================================================
#:for k1, t1 in REAL_KINDS_TYPES
elemental ${t1}$ function tanh_${k1}$( x ) result( y )
    ${t1}$, intent(in) :: x
    !==================================================
    y = tanh(x)
end function

elemental ${t1}$ function tanh_grad_${k1}$( x ) result( y )
    ${t1}$, intent(in) :: x
    !==================================================
    y = 1_${k1}$ - tanh(x)**2
end function

#:endfor

!==================================================
! Softmax
!==================================================
#:for k1, t1 in REAL_KINDS_TYPES
pure function Softmax_${k1}$( x ) result( y )
    ${t1}$, intent(in) :: x(:)
    ${t1}$ :: y(size(x))
    !==================================================
    y(:) = exp(x(:) - maxval(x(:)) )
    y(:) = y(:) / sum(y(:))
end function

pure function Softmax_grad_${k1}$( x ) result( y )
    ${t1}$, intent(in) :: x(:)
    ${t1}$ :: y(size(x))
    !==================================================
    y = softmax_${k1}$(x)
    y = y * (1_${k1}$ - y)
end function

#:endfor

!==================================================
! Softplus
!==================================================
#:for k1, t1 in REAL_KINDS_TYPES
elemental ${t1}$ function Softplus_${k1}$( x ) result( y )
    ${t1}$, intent(in) :: x
    !==================================================
    y = log(exp(x) + 1_${k1}$)
end function

elemental ${t1}$ function Softplus_grad_${k1}$( x ) result( y )
    ${t1}$, intent(in) :: x
    !==================================================
    y = exp(x) / (exp(x) + 1_${k1}$)
end function

#:endfor

end module

Some of them would be more interesting with the fast versions of some of the intrinsic functions. A companion stdlib_math_fast or stdlib_fast_math could be included.

[Beliavsky]

Since the GELU function is included in the code of @jalvesz, the approximations to it described at https://www.johndcook.com/blog/2025/03/06/gelu/ could also be considered for addition. Other posts by John D. Cook discussing activation functions are https://www.johndcook.com/blog/2023/08/06/swish-swiss/ and https://www.johndcook.com/blog/2023/08/06/swish-mish-and-serf/.

[jalvesz]

Can his formula be adopted in stdlib without licensing issues?

It seems it should not, as mathematical formulas (as opposed to code) are in principle not protected by of copy-right. If someone else could confirm, I can include this formula in the PR.