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Gaussian process (GP) emulator has been used as a surrogate model forpredicting force field and molecular potential, to overcome the computationalbottleneck of molecular dynamics simulation. Integrating both atomic force andenergy in predictions was found to be more accurate than using energy alone,yet it requires $O((NM)^3)$ computational operations for computing thelikelihood function and making predictions, where $N$ is the number of atomsand $M$ is the number of simulated configurations in the training sample, dueto the inversion of a large covariance matrix. The large computational needlimits its applications to emulating simulation of small molecules. Thecomputational challenge of using both gradient information and function valuesin GPs was recently noticed in statistics and machine learning communities,where conventional approximation methods, such as the low rank decomposition orsparse approximation, may not work well. Here we introduce a new approach, theatomized force field (AFF) model, that integrates both force and energy in theemulator with many fewer computational operations. The drastic reduction oncomputation is achieved by utilizing the naturally sparse structure of thecovariance satisfying the constraints of the energy conservation andpermutation symmetry of atoms. The efficient machine learning algorithm extendsthe limits of its applications on larger molecules under the same computationalbudget, with nearly no loss of predictive accuracy. Furthermore, our approachcontains uncertainty assessment of predictions of atomic forces and potentials,useful for developing a sequential design over the chemical input space, withalmost no increase in computational cost.

Efficient force field and energy emulation through partition of permutationally equivalent atoms