Fig. 1: On-the-fly training and acceleration of many-body Bayesian force fields (BFF).

a At each time-step of the MD simulation, local energies, forces, and stresses are computed with the SGP model. If the uncertainty on a local energy exceeds the chosen threshold ΔDFT, a DFT calculation is performed and the model is updated. b Mapping of local environments ρi onto multielement descriptors derived from the atomic cluster expansion. The environment is first mapped onto an equivariant descriptor ci, products of which are used to compute the rotationally invariant descriptor di that serves as an input to the model. c Mapping of a ξ = 2 SGP force field onto an equivalent quadratic model. The prediction cost of the SGP scales linearly with the number of sparse environments NS, while the cost of the corresponding polynomial model is independent of NS.