In GAP theory, a local energy is the portion of the total energy ascribed to an individual atom. While interactions based on physical models, like electrostatics or van der Waals, may have intuitive local energies associated to them, e.g., half of the pair-wise vdW interaction energy assigned to each atom in the pair, in general local GAP energies do not necessarily carry intuitive meaning. Formally, the local energies give the optimal piece-wise mathematical decomposition of the total energy, in the sense of reducing the error in the fit during training. In practice, local GAP energies give an idea of the contribution of the local atomic environment to the total cohesive energy of the system, and can in some sense be associated to the stability of a local atomic motif.
Currently, TurboGAP reports local energies in the first column of the energies.dat file. In the future, local energies will be reported in the trajectory_out.xyz file, as an array property following ASE's extended XYZ format.