Δ-HGPR: A Hierarchical Gaussian Process Framework for Active-Learning Δ-Potentials

Δ-learning, which uses machine learning to represent only the difference between a high-accuracy reference and a lower-accuracy base calculator, is a promising route to high-fidelity molecular dynamics (MD) with limited training data, but it has lacked a unified protocol combining hierarchical descriptors with well-defined equilibrium sampling. We propose Δ-HGPR (hierarchical Gaussian process regression for Δ-learning), a framework for learning difference potentials, and its MD instantiation, Δ-HGPR-MD. The framework combines (i) a difference potential written as an additive radial (inverse-distance) term plus a many-body machine-learning correction, determined by a single regularized least-squares fit; (ii) the ensemble force standard deviation as an active-learning indicator with an automatically scaled threshold; and (iii) a difference potential held fixed within each round so that equilibrium NVT sampling is well-defined in both exploration and production. The many-body term of Δ-HGPR-MD uses smooth overlap of atomic positions (SOAP) descriptors with random Fourier features (Δ-HGPR-RFF). On a toy system whose difference potential contains a 1/r contribution beyond the cutoff of the local descriptor, the many-body term alone reproduces none of this force, however many features it is given, whereas the hierarchy recovers 99% of it, with a reach set by the cutoff of its radial term. In MD, the hierarchy reproduces about 80% of the shift of the ion-pair distribution toward the reference, while the many-body term alone leaves it where the base places it. On malonaldehyde and the Zundel cation H5O2+ with three base calculators, Δ-HGPR-RFF removes barrier errors of up to 0.1 eV under one protocol, giving free-energy barriers statistically indistinguishable from DFT; free-energy perturbation shows that the error contributed by the learned correction is below 0.01 eV on average in every case. For H5O2+with ANI-2x, whose dynamics collapse without correction, it recovers the Zundel geometry and a barrier and transition count consistent with DFT.