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