Efficient parametrization of the atomic cluster expansion

The atomic cluster expansion (ACE) provides a general, local, and complete representation of atomic energies. Here we present an efficient framework for parametrization of ACE models for elements, alloys, and molecules. To this end, we first introduce general requirements for a physically meaningful description of the atomic interaction, in addition to the usual equivariance requirements. We then demonstrate that ACE can be converged systematically with respect to two fundamental characteristics—the number and complexity of basis functions and the choice of nonlinear representation. The construction of ACE parametrizations is illustrated for several representative examples with different bond chemistries, including metallic copper, covalent carbon, and several multicomponent molecular and alloy systems. We discuss the Pareto front of optimal force to energy matching contributions in the loss function, the influence of regularization, the importance of consistent and reliable reference data, and the necessity of unbiased validation. Our ACE parametrization strategy is implemented in the freely available software package pacemaker that enables largely automated and GPU accelerated training. The resulting ACE models are shown to be superior or comparable to the best currently available ML potentials and can be readily used in large-scale atomistic simulations.

Figure 1

Block scheme of the main pacemaker workflow.

Figure 2

Energy and force RMSE for train and test errors as a function of the relative energy and force contribution $\kappa$ to the loss function for the Cu-III dataset.

Figure 3

Relative energy and force RMSE for different ACE parametrizations for Cu, C, and ethanol. See text for details.

Figure 4

Energy and force RMSE for Cu-III test set trained using hierarchical body-order and power-order schemes in comparison to a single fit with the same basis functions. The values of $q$ were obtained from the force learning curve.

Figure 5

Energy vs nearest-neighbor distance curves for nonregularized, L1-regularized, and $w_0$-regularized potentials for fcc Cu (left panels) and diamond C (right panels) structures, trained on the three nested datasets of different sizes. The grey shaded regions in the graphs mark the energy ranges covered by each training dataset.

Figure 6

On-the-fly validation: variation of stacking fault energies in Cu during training [40]. ESF, ISF, and TWIN denote extrinsic, intrinsic, and twin stacking faults, respectively. Dashed lines mark the reference DFT values; the order of the ACE predicted energies matches that of DFT.

Figure 7

Error in the energy per atom as a function of the reference energy for the Cu-III test dataset. The red line shows the average error, the top panel the number of samples, and the right panel the overall error distribution, including a normal distribution with identical second moment.

Figure 8

Convergence of force components for ethanol train and test datasets. Horizontal lines indicate test results of ML potentials given in Ref. [72]. See text for details.

Figure 9

MAE for the enthalpy of formation versus the number of basis functions per element (only binary interactions were included in ACE). Horizontal lines indicate results of other ML methods investigated in Ref. [5] (upper pannel) and RMSE for the enthalpy of formation for individual binary alloys (lower panel).

Figure 10

ACE learning curve for the HEA dataset taken from Ref. [63].

Figure 11

pacemaker workflow scheme. See text.

Figure 12

Radial basis functions $g_k\left(r\right)$ available in pacemaker. (a) Exponentially-scaled Chebyshev polynomials with $\lambda =5.25$, (b) Power-law scaled Chebyshev polynomials with $\lambda =2.0$, (c) Simplified spherical Bessel functions. For all radial basis functions cutoff distance $r_{\mathrm{cut}}=5$.

Figure 13

LM-tree for $\nu =4$.

Figure 14

Dependencies of the energy per atom and force vectors RMSE on the L1 (top) and L2 (bottom) regularization parameters for train (solid lines) and test (dashed lines with crosses) sets of Cu-I, Cu-II, and Cu-III.

Figure 15

Extrapolation of ACE potential trained on Cu-I dataset to the prediction of Cu-II and Cu-III datasets.

Figure 16

Dependencies of the energy per atom and force vectors RMSE on the relative contribution of smoothness regularizations $w_0, w_1$ and $w_2$ to the total loss function for train (solid lines) and test (dashed lines) subsets of Cu-II and Cu-III.

Figure 17

Dependency of the computational time for the HEA ACE potential with respect to the number of basis functions per element (upper panel) and with respect to the number of atoms and number of cores (lower panel).