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

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.