We provide a definition and explicit expressions for $n$-body Gaussian process (GP) kernels, which can learn any interatomic interaction occurring in a physical system, up to $n$-body contributions, for any value of $n$. The series is complete, as it can be shown that the “universal approximator” squared exponential kernel can be written as a sum of $n$-body kernels. These recipes enable the choice of optimally efficient force models for each target system, as confirmed by extensive testing on various materials. We furthermore describe how the $n$-body kernels can be “mapped” on equivalent representations that provide database-size-independent predictions and are thus crucially more efficient. We explicitly carry out this mapping procedure for the first nontrivial (three-body) kernel of the series, and we show that this reproduces the GP-predicted forces with $\mathrm{meV}/\AA$ accuracy while being orders of magnitude faster. These results pave the way to using novel force models (here named “M-FFs”) that are computationally as fast as their corresponding standard parametrized $n$-body force fields, while retaining the nonparametric character, the ease of training and validation, and the accuracy of the best recently proposed machine-learning potentials.

• Received 15 January 2018
• Revised 24 April 2018

DOI:https://doi.org/10.1103/PhysRevB.97.184307