A Force-Kernel Reformulation of the Extended-System Adaptive Biasing Force for Free-Energy Calculations

We introduce force-kernel extended-system adaptive biasing force (FK-eABF), a force-based kernel reformulation of eABF that replaces the histogram-based mean-force accumulator of conventional eABF with a sparse population of Gaussian kernels storing local running-mean forces. Biasing forces are recovered by Nadaraya-Watson regression, yielding smooth estimates from the earliest stages of a simulation without a minimum-count threshold, while the same kernel population also defines an auxiliary, self-attenuating exploration force that requires no prior knowledge of barrier heights. On N-acetyl-N'-methylalanylamide in explicit water, FK-eABF achieves full free-energy landscape coverage faster than well-tempered metadynamics (WT-MetaD), on-the-fly probability enhanced sampling (OPES), and WTM-eABF, while all four methods converge to comparable accuracy given sufficient time. FK-eABF also retains long-time accuracy: on the DFG-in/out transition of Abl1 kinase, multimicrosecond simulations recover the established near-isoenergetic balance between states. At the opposite extreme, applied to the electrocyclic ring closure of 1,3-butadiene at the ab initio molecular dynamics level, FK-eABF recovers the free-energy landscape within 30 ps. Together, these benchmarks, spanning more than 4 orders of magnitude in simulation time, establish FK-eABF as more than a kernelized implementation of eABF: A force-based kernel reformulation that delivers faster early time convergence without sacrificing long-time quantitative accuracy.