Low-Scaling Algorithm for Nudged Elastic Band Calculations Using a Surrogate Machine Learning Model

We present the incorporation of a surrogate Gaussian process regression (GPR) atomistic model to greatly accelerate the rate of convergence of classical nudged elastic band (NEB) calculations. In our surrogate model approach, the cost of converging the elastic band no longer scales with the number of moving images on the path. This provides a far more efficient and robust transition state search. In contrast to a conventional NEB calculation, the algorithm presented here eliminates any need for manipulating the number of images to obtain a converged result. This is achieved by inventing a new convergence criteria that exploits the probabilistic nature of the GPR to use uncertainty estimates of all images in combination with the force in the saddle point in the target model potential. Our method is an order of magnitude faster in terms of function evaluations than the conventional NEB method with no accuracy loss for the converged energy barrier values.

Figure 1

Comparison between the (a) classical CI-NEB and (b) machine learning NEB (ML-NEB) methods. The performance of both algorithms is illustrated in the two-dimensional Müller-Brown PES. Predicted MEP is included at the bottom of each PES to show the evolution of the energy profile for the elastic band with respect to the number of function calls.

Figure 2

MEP for the (a) diffusion of a Au atom on an Al(111) surface, (b) diffusion of a Pt adatom on a stepped Pt surface across the two terraces and (c) rearrangement of a Pt heptamer island adsorbed on a Pt(111) surface obtained for the algorithms: FIRE, LBFGS, MDMin, and ML-NEB (using the three acquisition functions presented in the text). The number of function calls required for each algorithm to converge are shown in bold between brackets. The top and side views of the optimized initial and final states for each transition along with their corresponding saddle points are included at the top of each composition. The undercoordinated atoms of the Pt step edge in (b) are highlighted in blue.

Figure 3

Comparison of the number of function evaluations required to achieve convergence with an increasing number of images for the different classical and machine learning accelerated methods. The benchmark is performed with the classical method (using the FIRE, LBFGS, and MDMin algorithms) and the ML-NEB method (using the three acquisition functions described in the main text). The lower panels show the average error of the predicted energy along the path obtained by the three acquisition functions with respect to the target value of the function at the same geometric positions as the ones predicted by the ML-NEB.