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Atomic Cluster Expansion without Self-Interaction

Cheuk Hin Ho, Timon S. Gutleb, Christoph Ortner

TL;DR

This work addresses the tension in Atomic Cluster Expansion (ACE) between the physically motivated canonical expansion and the computationally efficient self-interacting variant. It introduces a purification operator that transforms the self-interacting basis into the canonical form, enabling efficient evaluation of the canonical expansion while preserving symmetry sparsification, and proves that the two expansions span comparable spaces under mild conditions. Theoretical results establish the existence and sparsity of the purification transform, its compatibility with $G$-symmetrization, and the construction of orthogonal symmetric bases; numerically, canonical ACE shows improved conditioning, coefficient decay alignment with optimal polynomial bases, and greater robustness under regularization. Empirically, canonical ACE provides better qualitative and quantitative performance in MLIPs on challenging datasets (Zuo et al. 2020 and Fe), supporting its adoption as a robust alternative to self-interacting ACE in symmetric, multi-body regression tasks.

Abstract

The Atomic Cluster Expansion (ACE) (Drautz, Phys. Rev. B 99, 2019) has been widely applied in high energy physics, quantum mechanics and atomistic modeling to construct many-body interaction models respecting physical symmetries. Computational efficiency is achieved by allowing non-physical self-interaction terms in the model. We propose and analyze an efficient method to evaluate and parameterize an orthogonal, or, non-self-interacting cluster expansion model. We present numerical experiments demonstrating improved conditioning and more robust approximation properties than the original expansion in regression tasks both in simplified toy problems and in applications in the machine learning of interatomic potentials.

Atomic Cluster Expansion without Self-Interaction

TL;DR

This work addresses the tension in Atomic Cluster Expansion (ACE) between the physically motivated canonical expansion and the computationally efficient self-interacting variant. It introduces a purification operator that transforms the self-interacting basis into the canonical form, enabling efficient evaluation of the canonical expansion while preserving symmetry sparsification, and proves that the two expansions span comparable spaces under mild conditions. Theoretical results establish the existence and sparsity of the purification transform, its compatibility with -symmetrization, and the construction of orthogonal symmetric bases; numerically, canonical ACE shows improved conditioning, coefficient decay alignment with optimal polynomial bases, and greater robustness under regularization. Empirically, canonical ACE provides better qualitative and quantitative performance in MLIPs on challenging datasets (Zuo et al. 2020 and Fe), supporting its adoption as a robust alternative to self-interacting ACE in symmetric, multi-body regression tasks.

Abstract

The Atomic Cluster Expansion (ACE) (Drautz, Phys. Rev. B 99, 2019) has been widely applied in high energy physics, quantum mechanics and atomistic modeling to construct many-body interaction models respecting physical symmetries. Computational efficiency is achieved by allowing non-physical self-interaction terms in the model. We propose and analyze an efficient method to evaluate and parameterize an orthogonal, or, non-self-interacting cluster expansion model. We present numerical experiments demonstrating improved conditioning and more robust approximation properties than the original expansion in regression tasks both in simplified toy problems and in applications in the machine learning of interatomic potentials.
Paper Structure (31 sections, 8 theorems, 69 equations, 11 figures, 3 tables)

This paper contains 31 sections, 8 theorems, 69 equations, 11 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Theory
  3. Two cluster expansions of point clouds
  4. Canonical cluster expansion
  5. Self-interacting cluster expansion
  6. Efficient Evaluation of the Canonical Cluster Expansion
  7. Implementation remarks
  8. Orthogonal Symmetric Basis
  9. Regularization of cluster expansions for multisets
  10. Purification and $G$-Symmetrization
  11. Review of $G$-symmetrization
  12. Effect of symmetrization for $G = O(1)$
  13. Effect of symmetrization for $G = SO(2)$
  14. Effect of symmetrization for $G = O(3)$
  15. Numerical Experiments: Academic Examples
  16. ...and 16 more sections

Key Result

Theorem 2.1

Let $\mathbf{K}$ be a finite ordered index set. Suppose that any pointwise product of $\phi_{k_1}(\mathbf{x})\phi_{k_2}(\mathbf{x})$ can be linearized exactly in terms of a finite sum of $\phi_{\kappa}(\mathbf{x})$eq:linearize_product. Then there exists a $\mathbf{K}' \supset \mathbf{K}$ such that $

Figures (11)

  • Figure 1: (a) Bound of non-zero terms in ${\bf k}'$ summation in \ref{["eq:linearizek'"]} when using a Chebyshev basis for embedding one-dimensional particles ${\bf x}_j \in [-1,1]$; The higher the correlation order and the lower the degree, the more the bound is overestimated. (b) Sparsity pattern of $\mathcal{P}$ for correlation order three and total degree = 20. Black pixels indicate non-zeros; sparsity $\approx 1.48\%$. $\mathcal{P}$ is triangular since the Chebyshev basis is total-degree preserving \ref{['equa:prodspan_totdeg']}.
  • Figure 2: Change of gram matrix condition numbers of the canonical and self-interacting bases with total degree $14$ for the indicated basis orders. A diagonal scaling is applied to the gram matrices consistent with Table \ref{['table:CondTable3D']}.
  • Figure 3: Coefficient decay in the canonical and self-interacting bases for $f_{25}$ as defined in \ref{['eq:testfunctioncoeffs1']}. All data was sampled independently from a tensor product of distributions with density $\frac{1}{\sqrt{1-x^2}}$ over $[-1,1]^N$ and regressed against an $N$--dimensional ACE basis. The coefficient decay estimate was obtained following lloydHypercube and is the same in (a) and (b).
  • Figure 4: RMSE when approximating the functions $f_{5}$ in \ref{['eq:testfunctioncoeffs1']} with the indicated basis functions of total degree 30 and differently distributed data.
  • Figure 5: Maximum error when approximating the functions $f_{5}$ in \ref{['eq:testfunctioncoeffs1']} with the indicated basis functions of total degree 30 using the uniform distribution with identical setting as in Figure \ref{['fig:cheblegendre+uniform+cheb']}.
  • ...and 6 more figures

Theorems & Definitions (16)

  • Theorem 2.1
  • Corollary 2.2
  • Remark 2.3
  • Proposition 2.4
  • Corollary 2.5
  • Proposition 2.6
  • Proposition 2.7
  • Proposition 2.8
  • proof
  • Lemma A.1
  • ...and 6 more