Delta-learned force fields for nonbonded interactions: Addressing the strength mismatch between covalent-nonbonded interaction for global models

Abstract

Noncovalent interactions--vdW dispersion, hydrogen/halogen bonding, ion-$π$, and $π$-stacking--govern structure, dynamics, and emergent phenomena in materials and molecular systems, yet accurately learning them alongside covalent forces remains a core challenge for machine-learned force fields (MLFFs). This challenge is acute for global models that use Coulomb-matrix (CM) descriptors compared under Euclidean/Frobenius metrics in multifragment settings. We show that the mismatch between predominantly covalent force labels and the CM's overrepresentation of intermolecular features biases single-model training and degrades force-field fidelity. To address this, we introduce \textit{$Δ$-sGDML}, a scale-aware formulation within the sGDML framework that explicitly decouples intra- and intermolecular physics by training fragment-specific models alongside a dedicated binding model, then composing them at inference. Across benzene dimers, host-guest complexes (C$_{60}$@buckycatcher, NO$_3^-$@i-corona[6]arene), benzene-water, and benzene-Na$^+$, \mbox{$Δ$-sGDML} delivers consistent gains over a single global model, with fragment-resolved force-error reductions up to \textbf{75\%}, without loss of energy accuracy. Furthermore, molecular-dynamics simulations further confirm that the $Δ$-model yields a reliable force field for C$_{60}$@buckycatcher, producing stable trajectories across a wide range of temperatures (10-400~K), unlike the single global model, which loses stability above $\sim$200~K. The method offers a practical route to homogenize per-fragment errors and recover reliable noncovalent physics in global MLFFs.