[Paper Review] Unified theory of atom-centered representations and message-passing machine-learning schemes

Jigyasa Nigam, Sergey N. Pozdnyakov+2 more|arXiv (Cornell University)|Feb 3, 2022

Machine Learning in Materials Science49 references42 citations

TL;DR

This paper unifies atom-centered and message-passing machine learning frameworks for atomistic materials modeling by generalizing atom-centered density correlations (ACDC) to multi-centered representations, providing a complete linear basis for symmetric, equivariant functions of atomic coordinates. The framework enables systematic construction of invariant and equivariant models with theoretical universality and practical performance across short- and long-range structure-property relationships.

ABSTRACT

Data-driven schemes that associate molecular and crystal structures with their microscopic properties share the need for a concise, effective description of the arrangement of their atomic constituents. Many types of models rely on descriptions of atom-centered environments, that are associated with an atomic property or with an atomic contribution to an extensive macroscopic quantity. Frameworks in this class can be understood in terms of atom-centered density correlations (ACDC), that are used as a basis for a body-ordered, symmetry-adapted expansion of the targets. Several other schemes, that gather information on the relationship between neighboring atoms using "message-passing" ideas, cannot be directly mapped to correlations centered around a single atom. We generalize the ACDC framework to include multi-centered information, generating representations that provide a complete linear basis to regress symmetric functions of atomic coordinates, and provides a coherent foundation to systematize our understanding of both atom-centered and message-passing, invariant and equivariant machine-learning schemes.

Motivation & Objective

• To unify atom-centered and message-passing machine learning schemes in atomistic modeling.
• To generalize atom-centered density correlations (ACDC) to include multi-centered, graph-based information.
• To provide a complete linear basis for symmetric and equivariant functions of atomic coordinates.
• To systematize understanding of both invariant and equivariant deep learning architectures in materials science.
• To enable performance gains in modeling short- and long-range structure-property relations.

Proposed method

• Generalizes ACDC to multi-centered representations using tensor products of neighbor environments and symmetry-adapted basis functions.
• Introduces message-passing ACDC representations via iterative summation over neighbors and higher-order contractions.
• Employs irreducible representations of SO(3) and parity labels to ensure rotational and inversion symmetry.
• Uses coupled-basis equivariants via Clebsch-Gordan decomposition to build higher-order symmetric features.
• Applies a density trick to avoid exponential scaling with neighbor count by operating on atom-centered densities.
• Employs radial and spherical harmonic basis functions to discretize continuous representations and enable efficient computation.

Experimental results

Research questions

• RQ1 How can atom-centered and message-passing frameworks be formally unified under a single theoretical framework?
• RQ2 What is the role of multi-centered correlations in constructing complete, symmetric, and equivariant representations?
• RQ3 Can the generalized ACDC formalism serve as a universal basis for both invariant and equivariant machine learning models?
• RQ4 How do message-passing representations compare to traditional ACDC in capturing long-range structural effects?
• RQ5 What is the theoretical foundation for the universality of these representations in approximating symmetric functions of atomic coordinates?

Key findings

• The generalized ACDC framework provides a complete linear basis for expanding any symmetric, equivariant function of atomic coordinates.
• Message-passing ACDC representations achieve high accuracy in modeling both short- and long-range structure-property relations.
• The framework unifies diverse models including graph neural networks, tensor field networks, and traditional ACDC schemes under a common formalism.
• Iterative coupling of equivariant features via Clebsch-Gordan rules enables richer, higher-order representations without loss of symmetry.
• The method avoids exponential scaling through a density-based trick, enabling efficient computation on large atomic environments.
• The approach is theoretically universal, equivalent to a complete basis of symmetric polynomials, and proven to support universal function approximation for equivariant models.