where 𝑁 ,-./"! , 𝐽, and 𝑆 denote the total number of geometries in adiabatic database, diabatic restraint database, and the permutation database, 𝛼 & and 𝛼 ' are weights employed for diabatic restraint and regularization, respectively, 𝐕 % , and 𝐕 ^ denote the target adiabatic potential energies for geometry n and s respectively, and 𝚯 [ denotes the known unique DPEM elements for geometry j in the diabatic restraint database.