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$\Delta$-Machine Learning ($\Delta$-ML) has been shown to effectively andefficiently bring a low-level ML potential energy surface to CCSD(T) quality.Here we propose extending this approach to general force fields, whichimplicitly or explicitly contain many-body effects. After describing thisgeneral approach, we illustrate it for the MB-pol water potential whichcontains CCSD(T) 2-body and 3-body interactions but relies on the TTM4-F 4-bodyand higher body interactions. The 4-body MB-pol (TTM4-F) interaction fails at avery short range and for the water hexamer errors up to 0.84 kcal/mol are seenfor some isomers, owing mainly to 4-body errors. We apply $\Delta$-ML for the4-body interaction, using a recent dataset of CCSD(T) 4-body energies that weused to develop a new water potential, q-AQUA. This 4-body correction is shownto improve the accuracy of the MB-pol potential for the relative energies of 8isomers of the water hexamer as well as the harmonic frequencies. The newpotential is robust in the very short range and so should be reliable forsimulations at high pressure and/or high temperature.

A $\Delta$-Machine Learning Approach for Force Fields, Illustrated by a CCSD(T) 4-body Correction to the MB-pol Water Potential