Note that an alternative to the above would be to constrain the permutation invariance of identical nuclei rather than to restrain it. This could be accomplished by using permutationally invariant input for input layer 𝐚 ) * (which constrains permutation invariance in the input stage), for example permutationally invariant polynomials or fundamental invariants, or by using an equivariant neural network that constrains permutation invariance in the output stage. Both approaches constrain the permutational invariance of adiabatic potential energies as well as DPEM elements. However, DPEM elements are not necessarily permutationally invariant, and the PR-DDNN method allows the combination of permutational variance of the diabatic representation with permutational invariance of the adiabatic one. Furthermore, the PR-DDNN method recognizes that one does not fit the unpermuted adiabatic data perfectly and allows imperfect fitting of permutational invariance, which is a balanced approach, rather than insisting that exact permutational invariance be combined with inexact fitting of adiabatic potentials.