Regularity Priors for the Linear Atomic Cluster Expansion

TL;DR

The paper argues that highly flexible ML interatomic potentials risk instability and poor extrapolation. It introduces regularity priors within the linear ACE framework to bias the PES toward smoothness, interpreting these priors as a rescaling of basis functions and as modifications to the neighbor density. Through extensive tests on Silicon (Si-GAP-18 and Si10pc) and Aspirin, the authors show that Gaussian-type regularity priors substantially improve force and energy errors and markedly enhance MD stability and PES smoothness, with gains persisting across data-rich and extrapolative regimes. The results suggest regularity priors are a practical, cost-free improvement for ACE and potentially for non-linear architectures, offering a path toward more reliable, extrapolatable MLIPs.

Abstract

Machine-learned interatomic potentials enable large systems to be simulated for long time scales at near ab-initio accuracy. This accuracy is achieved by fitting extremely flexible model architectures to high quality reference data. In practice, this flexibility can cause unwanted behavior such as jagged predicted potential energy surfaces and generally poor out-of-distribution behavior. We investigate a general strategy for incorporating prior beliefs on the regularity of the target energy into linear ACE models and explore to what extent this approach improves the quality of the fitted models. Our main focus is an over-regularisation that replicates the Gaussian broadening used in SOAP descriptors within the ACE framework.

Figures (28)

• Figure 1: The energy and force predictions on the Mg-Mg dimer are shown for various foundation models batatia2023foundationyang2024mattersimneumann2024orbpark2024scalablefu2025learningbarroso2024open as a function of dimer separation. The MLIP Arena python package chiang2025mlip was used to load all models and obtain the reference PBE result, originally computed with VASP kresse1996efficiency. For the avoidance of doubt model versions are MACE-MPA-0, MatterSim-v1.0.0-5M, ORB-v2, 7net-0, eSEN-30M-OAM and eqV2_86M_omat_mp_salex.
• Figure 2: The neighbor density $\tilde{\rho}$ corresponding to different choices of regularity prior are shown for a single neighbor atom at r=(3,0,0). The left-hand column shows a radial cut along $(r, 0, 0)$ while the right-hand column shows a perpendicular cut along $(3, r, 0)$. The black line shows the density with no regularity prior (a delta function represented using $n_\mathrm{max}=l_\mathrm{max}=20$) whilst the legend indicates the regularity parameter in $\lambda_{nl}^{-p/2}$, $\exp{\left(-\alpha\sqrt{\lambda_{nl}}\right)}$ and $\exp\left(-\sigma^2/2 \lambda_{nl} \right)$ for the polynomial, exponential and Gaussian priors respectively, where $\lambda_{nl}$ are the eigenvalues of $\Delta$. To aid visual comparison the maximum value of $\tilde{\rho}$ has been scaled to 1 for all curves. The black circles indicate the cutoff radius. The oscillatory behavior of the densities for low smoothing occur naturally in high-accuracy approximations of a delta function (cf. Dirichlet kernel and analogy with Methfessel-Paxton smearing methfessel1989high) and may be beneficial in the context of very dense datasets.
• Figure 3: Force and energy errors on configurations in the training and test sets, as well as two different configuration types from ref. morrow2022indirect, see main text for details, not present in the training data, are shown for ACE models fit using Gaussian regularity priors. Larger values of sigma correspond to a more regular prior. The force error on the training set, shown in the legend, is constant as this was used to select the strength of the regularization.
• Figure 4: Force RMSE on the training and test sets is shown for potential fit to Si10pc using a Gaussian regularity prior with varying $\sigma$. Each marker corresponds to a particular choice of Tikhonov regularization strength.
• Figure 5: The stress (derivative of energy) is plotted as a function of separation for decohesion of silicon diamond into the 100, 110 and 111 surfaces. The curves for different values of $\sigma$ are offset (black arrows) to aid comparison. The bottom right panel shows the energy of the Si-Si dimer as a function of separation between the atoms.