[Paper Review] Completeness of Atomic Structure Representations

Jigyasa Nigam, Sergey N. Pozdnyakov+2 more|arXiv (Cornell University)|Feb 28, 2023

Machine Learning in Materials ScienceMaterials Science56 references3 citations

TL;DR

This paper proposes a novel symmetry-adapted descriptor based on finite-order three-body correlations to achieve completeness in atomic structure representations, overcoming limitations of conventional density-correlation methods. The method enables universal approximation of atomic properties with convergence controlled solely by neighbor discretization resolution, demonstrated by successfully distinguishing degenerate configurations that existing models fail to resolve.

ABSTRACT

In this paper, we address the challenge of obtaining a comprehensive and symmetric representation of point particle groups, such as atoms in a molecule, which is crucial in physics and theoretical chemistry. The problem has become even more important with the widespread adoption of machine-learning techniques in science, as it underpins the capacity of models to accurately reproduce physical relationships while being consistent with fundamental symmetries and conservation laws. However, some of the descriptors that are commonly used to represent point clouds -- most notably those based on discretized correlations of the neighbor density, that underpin most of the existing ML models of matter at the atomic scale -- are unable to distinguish between special arrangements of particles in three dimensions. This makes it impossible to machine learn their properties. Atom-density correlations are provably complete in the limit in which they simultaneously describe the mutual relationship between all atoms, which is impractical. We present a novel approach to construct descriptors of \emph{finite} correlations based on the relative arrangement of particle triplets, which can be employed to create symmetry-adapted models with universal approximation capabilities, which have the resolution of the neighbor discretization as the sole convergence parameter. Our strategy is demonstrated on a class of atomic arrangements that are specifically built to defy a broad class of conventional symmetric descriptors, showcasing its potential for addressing their limitations.

Motivation & Objective

• To address the incompleteness of widely used atomic structure descriptors based on neighbor density correlations, which fail to distinguish non-symmetrically related atomic arrangements.
• To develop a symmetry-adapted, finite-order descriptor that ensures completeness in representing atomic environments while maintaining computational efficiency.
• To enable universal approximation of atomic properties by constructing a complete basis set from finite correlations of particle triplets.
• To demonstrate the method's ability to resolve degenerate configurations that conventional models cannot distinguish, even under small perturbations.
• To provide a scalable, convergence-controlled alternative to deep neural networks for machine learning in atomistic modeling.

Proposed method

• The method constructs invariants from relative arrangements of particle triplets, forming a complete basis for symmetry-adapted representations of atomic environments.
• It uses finite-order correlations (specifically, three-body correlations) to define descriptors that are provably complete under the condition of finite neighbor discretization.
• The descriptors are built as symmetric functions of triplet configurations, ensuring invariance under global translations, rotations, and permutations of equivalent atoms.
• The approach leverages a hierarchical encoding of triplet correlations to form a universal approximator with convergence governed only by the resolution of neighbor discretization.
• The framework is validated using a dataset of degenerate B8 clusters, comparing performance across linear and nonlinear models trained on different feature sets.
• A two-stage training pipeline is employed: first learning high-order features (e.g., $ ho_i^{igotimes 7}$) from triplet descriptors, then predicting energies from these features.

Experimental results

Research questions

• RQ1 Can a finite-order correlation descriptor based on particle triplets achieve completeness in representing atomic structures, distinguishing all non-symmetrically equivalent configurations?
• RQ2 How does the proposed descriptor compare to conventional density-correlation-based descriptors in resolving degenerate atomic arrangements?
• RQ3 To what extent can the descriptor support universal approximation of atomic properties with controlled convergence via discretization resolution?
• RQ4 Does the method maintain numerical stability and accuracy when applied to highly symmetric, degenerate clusters such as B8?
• RQ5 Can the descriptor be efficiently implemented in data-driven models without requiring infinitely deep networks?

Key findings

• The proposed triplet-based descriptor achieves completeness in representing atomic structures, successfully distinguishing configurations that are degenerate under conventional one- and two-body descriptors.
• The method enables universal approximation of atomic properties with convergence controlled solely by the resolution of neighbor discretization, eliminating the need for infinite network depth.
• Training on the triplet descriptors reduced the relative error on the energy difference between degenerate B8 pairs to 0.0481 for nonlinear models (A$_{\text{NL}}$), compared to 1.00 for models using only $\nu=1$ features (B$_{\text{NL}}$).
• The nonlinear model trained on high-order features ($\rho_i^{igotimes 7}$) achieved a relative training error of 0.001 on the energy of degenerate pairs, demonstrating high accuracy and stability.
• The method outperforms conventional approaches in resolving structural degeneracy, with a relative error on the difference of features between degenerate pairs reduced to 0.125 for the best-performing model (A$_{\text{NL}}^\rho$).
• The results confirm that triplet correlations provide a complete, scalable, and efficient alternative to traditional density-correlation methods in machine learning for atomistic systems.