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Atomic Cluster Expansion: Completeness, Efficiency and Stability

Genevieve Dusson, Markus Bachmayr, Gabor Csanyi, Ralf Drautz, Simon Etter, Cas van der Oord, Christoph Ortner

arXiv:1911.03550v4math.NA

TL;DR

The paper addresses how to construct complete, efficient polynomial bases for approximating isometry- and permutation-invariant atomistic functions. It develops ACE basis construction and numerical evaluation methods, showing that careful construction yields a complete basis that is highly efficient to evaluate, while identifying inverse-problem and linear-dependence challenges.

  • Problem

    A systematic basis is needed for approximating symmetric atomistic functions while preserving isometry and permutation invariance and supporting numerical analysis.

  • Method

    The paper derives ACE using symmetric polynomials and spherical harmonics, adds an offline basis-construction algorithm, and develops recursive and adjoint-based evaluation procedures.

  • Results

    Careful construction yields a complete basis that is highly efficient to evaluate.

  • Takeaways & Limitations

    The resulting basis provides a foundation for studying approximation properties and parameter-estimation schemes for symmetric atomistic models.

  • Takeaways & Limitations

    The inverse problem of uniquely determining interaction terms from condensed-state training data remains unresolved and can be ill-conditioned.

Abstract

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The Atomic Cluster Expansion (Drautz, Phys. Rev. B 99, 2019) provides a framework to systematically derive polynomial basis functions for approximating isometry and permutation invariant functions, particularly with an eye to modelling properties of atomistic systems. Our presentation extends the derivation by proposing a precomputation algorithm that yields immediate guarantees that a complete basis is obtained. We provide a fast recursive algorithm for efficient evaluation and illustrate its performance in numerical tests. Finally, we discuss generalisations and open challenges, particularly from a numerical stability perspective, around basis optimisation and parameter estimation, paving the way towards a comprehensive analysis of the convergence to a high-fidelity reference model.

1. Introduction

The paper targets approximation of functions invariant under isometries and permutations, especially many-particle models, and presents ACE as a systematic symmetric-polynomial framework. It develops complete-basis construction, efficient evaluation, and numerical-analysis directions while leaving practical interatomic-potential applications for future work.

  • ACE systematically constructs symmetric polynomial representations using spherical harmonics for functions invariant under isometries and permutations.
  • The paper provides an offline algorithm to finalize the construction and obtain a complete basis.
  • A directed-acyclic-graph representation reduces ACE evaluation cost, with convergence tests on Silicon and Tungsten illustrating computational performance.
  • The paper examines parameter-estimation well-posedness, regularization, basis ill-conditioning, orthogonalization, and extensions to nonlinear regression and other symmetries.
  • Practical interatomic-potential applications are deferred to separate works, so this note is primarily a first step toward numerical analysis.

2. Potential Energy Surfaces

The paper models potential energy surfaces through body-order functions that satisfy regularity, locality, isometry invariance, and permutation invariance. It constructs polynomial approximations from radial functions and spherical harmonics, then combines them into a basis capable of approximating admissible N-body potentials.

  • Potential energy surfaces are represented as sums of site energies and N-body functions depending on neighbor positions relative to a center atom.
  • The assumed N-body potentials are regular, local within a cutoff, invariant under isometries, and invariant under permutations.
  • The formulation excludes collision configurations through a fixed minimum distance r0 and imposes cutoff-based support restrictions.
  • Spherical harmonics provide the angular component and enable explicit incorporation of O(3)-symmetry, while radial bases remain flexible.
  • The tensor-product basis can approximate any admissible N-body potential under the stated assumptions on the potential and radial basis.

3. Symmetric Polynomials

The paper constructs polynomial bases that preserve permutation and isometry symmetries, using spherical harmonics and explicit or numerical linear-algebra procedures. It establishes spanning and orthogonality properties, reduces basis sizes through symmetry, and identifies linear-dependence and stability challenges.

  • Symmetry-preserving approximation: Symmetrising an approximation preserves or improves its accuracy, allowing the approximant to inherit the target function’s permutation and isometry symmetries.The construction applies permutation, reflection, and rotation symmetries without increasing approximation error.
  • Symmetry-preserving approximation: The general construction averages tensor-product functions over the combined permutation and isometry group, then extracts a basis from the resulting symmetrised functions.The symmetrised set may be linearly dependent, so basis extraction is a separate computational step.
  • Rotation invariance: Spherical harmonics make rotation integration explicit and yield rotation-invariant functions by coupling angular momenta to zero total angular momentum.The resulting rotation-invariant functions form an orthonormal basis of the rotation-invariant subspace described in the paper.
  • Rotation invariance: The paper presents both a straightforward semi-numerical construction and an explicit SVD-based construction of rotation-invariant bases.The SVD construction exposes orthogonality and supports estimating the number of basis functions, while the semi-numerical approach is easy to implement and generalise.
  • Combined invariance: Combining permutation and rotation invariance can substantially reduce basis size, while distinct-index cases permit matching counts of permutation-invariant and rotation-permutation-invariant functions.The paper gives representative reductions in Table 1 and a proposition for pairwise-distinct index tuples.
  • Combined invariance: When indices are repeated, symmetries of the generalised Clebsch–Gordan coefficients create additional linear dependence, especially at low degree and interaction order N ≥4.Unified explicit formulas for all N are not yet available, motivating further work on coefficient symmetries or combined group representations.

4. Efficient Evaluation

The section develops efficient representations and evaluation algorithms for the ACE basis, replacing factorial permutation sums with correlation-based and recursive constructions. Symmetry-adapted correlations form a complete basis of symmetric polynomials, while directed-acyclic-graph evaluation can reduce arithmetic cost, subject to auxiliary basis functions for nonsplittable terms.

  • Correlation-based evaluation: The alternative basis avoids the N! cost of symmetrizing over permutations.Its basis functions are obtained from correlations, with subsequent conversion to the orthogonal basis through a sparse matrix-vector operation.
  • Completeness: The symmetry-adapted N-correlations form a complete basis of symmetric polynomials.For a radial basis satisfying the stated assumption, the resulting basis is also linearly independent.
  • Correlation-based evaluation: The additional evaluation cost scales linearly with the number of basis functions and the body-order.This yields an efficient representation of the space of symmetric polynomials.
  • Recursive representation: A directed acyclic graph can represent multiplicative basis decompositions, requiring one arithmetic operation per basis function when every node splits recursively.The performance improvement becomes increasingly significant with increasing interaction order.
  • Recursive representation: Some basis functions cannot be split while preserving the even-angular-order and zero-magnetic-index conditions, so auxiliary functions extend the basis.The extended set is non-unique, and optimizing it for minimal evaluation complexity remains open.
  • Recursive representation: The original explicitly orthogonal basis can still be evaluated through the extended representation using a sparse multiplication with additional empty rows.The extended evaluation targets the permutation-invariant basis without changing the original basis algorithmically.

5. Polynomial Radial Bases

The section constructs flexible radial bases by transforming the radial coordinate, applying a cutoff, and orthogonalizing polynomials under a user-defined measure. The resulting bases are orthogonal and support several optimization choices, although their performance and transferability require further study.

  • Construction: The proposed procedure constructs radial bases that are orthogonal under a user-defined inner product.It combines a domain, coordinate transformation, cutoff function, and orthogonality measure.
  • Orthogonal polynomials: The transformed polynomials can be evaluated by a recurrence relation that provides a fast and numerically stable means of evaluation.The recurrence coefficients are computable when polynomial integrals under the transformed measure are available.
  • Properties: The resulting radial basis satisfies the required assumption and is orthogonal in L2(ρ).The orthogonality properties follow by construction, while the polynomial span is dense in the target continuous function space.
  • Design choices: The cutoff power p = 2 is canonical because it is the lowest power giving Ct-regularity across the cutoff.The construction remains flexible in the choice of cutoff function and transformed measure.
  • Design choices: Independent inner and outer cutoffs can support a multiscale basis for modelling long-range interactions.The inner cutoff also prevents basis functions from oscillating or diverging below r0.
  • Open challenges: The radial basis offers substantial freedom for optimization, but systematic performance gains and possible transferability loss remain unresolved.Potential choices include nonlinear transformation optimization, angular-momentum-dependent bases, and independent cutoff parameters.

6. Parameter Estimation

The section formulates ACE parameter estimation from energy and force observations as a weighted linear least-squares problem. It establishes convergence under explicit basis, data, separation, consistency, and weight assumptions, while noting that regularization and electronic-structure consistency remain important limitations.

  • Model and data: A finite symmetric basis and its coefficients define a site potential and a parameterized potential energy surface.The section then studies estimation of the coefficient vector from configuration observations.
  • Estimation: The training data may contain total energies, forces, virials, force constants, and other observations generated by a reference model.The objective uses weights that can depend on configurations and observations.
  • Estimation: Energy and force fitting is a linear least-squares problem because both quantities are linear functionals of the parameters.The resulting system can be solved using QR or rank-revealing QR factorization.
  • Limitations: The least-squares formulation should include regularization because the inverse problem can be ill-posed.The section omits the regularization term in its main convergence-focused formulation and discusses it separately.
  • Convergence: Convergence of the optimized RMSE follows under the stated basis completeness, data regularity, uniform separation, model consistency, and weight-scaling conditions.The proof uses uniform convergence of potentials and gradients together with a uniform bound on local contributions.
  • Limitations: Model consistency is a severe restriction because electronic-structure observations typically do not satisfy it.A generalized result may require increasing interaction order and cutoff radius as the approximation sequence grows.

7. Implementation and Performance

The section specifies how to construct, evaluate, fit, and optimize ACE bases and site potentials. Recursive evaluation improves performance, while accuracy depends on interaction order, polynomial degree, cutoff, and training set.

  • Basis construction: A symmetric polynomial basis is specified by choosing radial functions, selecting finite index tuples, and constructing permutation-symmetry coupling coefficients.Degree functions can include tensor, total, or weighted degree, with interaction order determined by tuple length.
  • Evaluation: The evaluation pipeline identifies neighbors within rcut, evaluates one-particle and permutation-invariant many-body bases, and assembles the RPI basis.The many-body basis can use either the naive formula or the recursive representation.
  • Potential representation: The final site-potential representation is unique because permutation symmetry is carried by invariant basis functions and O(3) symmetry by the coefficients.After coefficients are fitted, the potential can be evaluated without separately constructing the RPI basis because symmetry is encoded in the coefficients.
  • Convergence tests: 1.43 meV/atom is achieved for W at N=3 with 1.12 ms/atom force evaluation, while N=5 gives 1.55 meV/atom at 0.55 ms/atom.At fixed basis-function count, increasing interaction order trades off against polynomial degree, so RMSE need not decrease monotonically.
  • Performance: The recursive evaluator improves performance by roughly an order of magnitude in the realistic setting, although practical speedups depend on one-particle-basis costs.Reported timings are competitive with similarly rich machine-learning potentials and do not appear to scale with interaction order N.
  • Cutoff dependence: Increasing basis size can reverse the preferred cutoff: smaller rcut performs better for relatively small bases, whereas larger rcut improves accuracy for larger bases.The comparison varies rcut across 4.0 Å, 5.0 Å, and 6.0 Å for W and Si.

8. Variations, Extensions and Remarks

The paper extends ACE through nonlinear compositions, recursive orthogonal-basis evaluation, and analysis of inverse-problem uniqueness. These extensions expose a trade-off between computational efficiency and numerical conditioning.

  • Generalizations: Nonlinear combinations and compositions of atom-centered symmetric features expand the design space beyond linear symmetric-polynomial site potentials.Features may be analytic, fitted to training data, vectorial, or defined self-consistently through a fixed-point map.
  • Basis conditioning: The efficient B_N basis introduces unphysical self-interactions, so changing order-N coefficients may require compensating changes at lower orders.This coupling indicates ill-conditioning with respect to the canonical inner product and complicates regularization.
  • Basis conditioning: The canonical symmetrized basis preserves pure interaction order and orthogonality, but its explicit permutation and cluster sums make it expensive to evaluate.This motivates seeking orthogonal bases with improved computational cost.
  • Recursive evaluation: The recursion provides an efficient route from the B_N basis to alternative basis functions because both bases are complete.The paper presents the recursion as potentially sufficient for training-phase basis evaluation, while the fitted potential uses a different representation.
  • Basis conditioning: For the pure B_N basis, the Gramian condition number is κ(G)=1, whereas combining interaction orders produces ill-conditioning from self-interactions.Table 2 estimates Gramian condition numbers using random samples under the canonical inner product.
  • Inverse problem: Whether condensed-state training data uniquely determines the potential remains unresolved, even when the potential is uniquely determined in principle.The paper leaves this inverse problem for future work and notes that related inversion results can be ill-conditioned.

9. Conclusions and Outlook

The paper presents ACE as a complete and efficiently evaluable basis for isometry- and permutation-invariant atomistic functions. It positions this construction as a foundation for future analysis of approximation and parameter estimation, while practical potential development remains outside its scope.

  • Conclusions: The paper provides a detailed construction of polynomial approximations for isometry- and permutation-invariant atomistic models.Its intended audience includes mathematicians and numerical analysts entering the machine-learning interatomic-potential field.
  • Conclusions: With appropriate care, the construction yields a complete basis that is highly efficient to evaluate.The authors identify this as a foundation for analyzing approximation properties and parameter-estimation schemes.
  • Outlook: The paper identifies basis construction, theoretical foundations, and parameter estimation as areas requiring further numerical analysis.The conclusion frames these as challenges for improving the approach rather than as completed results.

A.1. Numerically stable evaluation of Y m

This appendix addresses numerical stability when evaluating spherical harmonics and their gradients. It uses recursive evaluation and removes the apparent pole singularity in gradient formulas.

  • Stable evaluation: The implementation evaluates spherical harmonics with methods adapted from prior work and additionally requires numerically stable gradients.The gradient treatment needs extra care beyond evaluating the harmonics themselves.
  • Stable evaluation: A recursion is used to evaluate the associated Legendre functions underlying the spherical harmonics.The implementation precomputes the relevant quantities and follows the stated recursive procedure.
  • Pole treatment: At the poles, where sin θ=0, the apparent (sin θ)^−1 singularity in the analytic gradient expression is removable.The appendix treats separate cases, including m=0 and m≠0, to eliminate the singular contribution.

A.2. Recursion for the Dl µm(Q) coefficients.

The appendix recursively computes Wigner D-matrix coefficients, building higher-order coefficients from previously computed lower-order values. It also connects the construction to basis dimensions and broader representation choices.

  • Recursion for the D coefficients: The coefficients D^l_{mμ}(Q) are computed recursively by assuming all coefficients with l ∈ N^{N−1} are already available.The recursion then constructs coefficients for l ∈ N^N using the product formula for D-matrices.
  • Recursion for the D coefficients: The resulting recursion formula expresses higher-order coefficients through intermediate angular-momentum indices and Clebsch–Gordan coefficients.The displayed terms include intermediate labels such as L, l′, and M̃.
  • Basis dimensions: For interaction orders N = 4 and 5, the appendix reports dimensions for different l and corresponding RPI bases for different n.For N ≤ 3, the dimension is always zero or one.
  • Related representations: The construction is presented alongside alternative symmetric-polynomial bases and descriptors whose representation affects stability, conditioning, and evaluation cost.The comparison includes SOAP and related bases.

B.1. Atom-centred permutation-invariant potentials (aPIPs).

aPIPs constructs atom-centred permutation-invariant potentials from invariant coordinates and primary and secondary polynomial invariants. Its finite bases can span the same space as the ACE basis, but orthogonalization, computational scaling, and automated generation become challenging at higher body order.

  • Construction: aPIPs begins by choosing rotation-invariant coordinates, such as distances or distances together with angles.The angle-based discussion does not apply when only distance coordinates are used.
  • Construction: Invariant theory provides primary invariants I_1, …, I_dN and secondary invariants J_1, …, J_M for representing permutation-invariant polynomials.Any polynomial is written using secondary invariants multiplied by polynomials in the primary invariants.
  • Relation to ACE: With a suitable basis choice for the polynomial factors, a finite aPIP basis spans the same space as the analogous ACE basis.The equivalence is stated for the symmetric-polynomial spaces represented by the two constructions.
  • Limitations: aPIPs appears harder to orthogonalize than the spherical-harmonics construction and can have poor scaling with interaction order and neighbour count.The scaling may be more favorable for multiple species or open structures.
  • Limitations: Computer-algebra generation is currently unable to construct a complete same-species aPIPs basis for a 6-body potential with five neighbours.The paper notes that its constructions may generate these invariants, although this had not yet been explored.

B.2. Moment Tensor Potential (MTP).

MTP represents local environments with rotationally covariant moment tensors and contracts them into invariant basis functions. The paper gives an explicit connection to ACE, while noting sparse representations and unresolved numerical dependence issues in both approaches.

  • MTP construction: MTP uses rotationally covariant moment tensors derived from projections of the atomic density onto tensor bases.These moment tensors are subsequently contracted into rotationally invariant scalars.
  • MTP construction: The tensor contractions produce rotationally invariant scalar basis functions for a linear moment tensor potential expansion.More general contractions involving multiple tensors are also possible.
  • Numerical issues: Combining rotation and permutation invariance can introduce linear dependence, requiring numerical resolution; ACE can reduce the basis in small blocks, possibly symbolically.The paper contrasts this with MTP regularisation strategies that reduce the need for linear independence.
  • Representation and efficiency: Expanding tensor products in spherical harmonics yields a sparse representation of moment tensors, with (ν + 1)^2 spherical harmonics versus 3^ν matrix elements.Because MTP also uses sparse storage, the relative efficiency of the two implementations is difficult to compare without further study.
  • Relation to ACE: Replacing MTP moment tensors by the paper’s ACE expansion and ordering terms by powers of A provides an explicit basis connection.The paper concludes that suitable finite MTP and ACE sets span the same space and allow theoretical results to transfer between them.

B.3. SOAP.

SOAP uses a rotation- and permutation-invariant descriptor based on low-order correlations, while ACE generalizes the symmetric-polynomial construction to arbitrary interaction orders. Higher orders may distinguish more configurations, but complete injectivity is not established.

  • SOAP representation: SOAP regularises its descriptor in practice by replacing delta distributions with Gaussians, supporting accurate and transferable interatomic potentials.The cited applications use this descriptor for a variety of interatomic-potential constructions.
  • Injectivity: The SOAP descriptor X^(2) is not injective because it accounts only for 3-body correlations.Distinct atomic environments can therefore share the same descriptor even after accounting for symmetries.
  • ACE generalisation: A finite ACE basis containing sufficiently rich higher-order terms defines a descriptor that extends beyond the 3-body correlation setting.The construction permits interaction orders N > 2.
  • ACE generalisation: The symmetric-polynomial construction generalizes SOAP to arbitrary interaction orders.The paper presents this as a possible route toward distinguishing all configurations in a given training set.
  • Related descriptors: Hyperspherical-harmonic SOAP variants introduce 4-body correlations, and SNAP can be represented exactly in ACE form using linear regression.The mapping expresses hyperspherical harmonics as radial functions times spherical harmonics.
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