Optimization of the weights and biases according to eq (4) does not ensure that the adiabatic energies are invariant to permutation of identical nuclei. To promote permutational invariance, we add an additional database called permuted adiabatic database 𝒮. Database 𝒮 has 𝑆 = 𝑁 8 𝑁 geometries, where 𝑁 8 is the number of non-identity permutations of identical nuclei. For example, in O3, there are three transpositions and two cyclic permutations of O atoms, and hence 𝑁 8 = 5. For each geometry s in the permuted adiabatic database 𝒮, the target adiabatic potential energies 𝐕 9 are known because they are identical to those for the unpermuted geometries in database 𝒩.