program test
implicit none
integer ( kind = 4 ), parameter :: Nx = 64, Ny = 64
real ( kind = 8 ) , dimension ( Nx, Ny ) :: phi, dfdphi
integer ( kind = 4 ) :: i, j
call random_number (phi)
do concurrent ( i = 1 : Nx , j = 1 : Ny )
call Get_dphi ( phi, dfdphi, i, j )
end do
end program test

I think gfortran cannot see that the subroutine is pure because it is not in a module and an INTERFACE was not provided. Gfortran 13.0.0 20221218 compiles the following code, in which the subroutine is put in a module:

module m
contains
pure subroutine Get_dphi ( phi_, dfdphi_, i_, j_ )
implicit none
integer ( kind = 4 ), parameter :: Nx_ = 64,Ny_ = 64
real ( kind = 8 ), dimension ( Nx_, Ny_ ), intent ( in ) :: phi_
real ( kind = 8 ), dimension ( Nx_, Ny_ ), intent ( out ) :: dfdphi_
integer ( kind = 4 ), intent ( in ) :: i_, j_
dfdphi_(i_,j_) = 2.0*phi_(i_,j_)*( 1.0 - phi_(i_,j_) )**2 &
*( 1.0 - 2*phi_(i_,j_) )
end subroutine Get_dphi
end module m
program test
use m
implicit none
integer ( kind = 4 ), parameter :: Nx = 64, Ny = 64
real ( kind = 8 ) , dimension ( Nx, Ny ) :: phi, dfdphi
integer ( kind = 4 ) :: i, j
call random_number (phi)
do concurrent ( i = 1 : Nx , j = 1 : Ny )
call Get_dphi ( phi, dfdphi, i, j )
end do
end program test

As illustrated by @Beliavsky , a reference to a PURE subprogram (e.g., in your main program) requires an explicit interface.

Try to avoid using implicit interfaces as much as you can. Consider using an explicit interface as imperative with all features introduced in the language starting Fortran 90; PURE procedures a good example of this.

By the way, you may want to consider moving away from â€śmagic numbersâ€ť such as 8 for the KINDs of intrinsic types, instead try using named constants with language intrinsics such as SELECTED_REAL_KIND.