Graph Atomic Cluster Expansion for semilocal interactions beyond equivariant message passing

I. INTRODUCTION

• Graph ACE extends local ACE to represent semilocal interactions and encompasses atom-centered local ACE and multi-atom, multi-layer message-passing architectures.The paper presents this as a generalization that makes relations among ACE variants more transparent.Source: §I · pp. 1–2

II. GRAPH ATOMIC CLUSTER EXPANSION

• ACE(g) avoids assuming an atomic energy decomposition from the outset, with ACE(l) forming a subset of the global graph expansion.Local ACE decomposes energy into atomic contributions and limits direct pairwise interactions to a cutoff distance.Source: §II · p. 2

A. Configurations

• ACE(g) includes admissible configurations beyond the atom-centered configurations of ACE(l), enabling configurations suited to semilocal interactions.Each admissible configuration must allow atomic positions to be reconstructed up to a translation.Source: A. Configurations · pp. 2–3, Figure 1 · p. 3

B. Single-particle basis functions

• The single-particle basis uses atom-centered functions of position and atomic variables, with orthonormality and completeness assumed for these bases.The positional functions approach zero at a cutoff distance; species- or variable-dependent functions can represent discrete or continuous atomic properties.Source: B. Single-particle basis functions · p. 3

C. Cluster basis functions

• Cluster basis functions are products of single-particle basis functions, and each configuration provides a complete, orthonormal basis for the system properties considered.Using all configurations produces a globally overcomplete set, which the paper says allows selection of sensitive basis functions.Source: C. Cluster basis functions · pp. 3–4

D. Graph structure and reduction

• Cluster graphs encode directed bonds with positional basis functions on edges and atomic-variable basis functions on nodes; topology classifies their basis functions and coefficients.For a given configuration, coefficients are independent of atomic positions.Source: D. Graph structure and reduction · p. 4, §1 · p. 4

2. Connected graphs

• A graph basis function contributes to local properties only when every node is reachable from every other node along graph edges, regardless of edge orientation.Disconnected graph fragments can be separated without changing the function's value, so they do not describe local interactions between the fragments.Source: §2 · p. 4, Figure 2 · p. 4

3. Edges shorter than cut-off distance

• A graph basis function with an edge longer than rcut vanishes identically and cannot contribute.The figure contrasts the admissible connected graph with disconnected and over-cutoff examples.Source: §3 · p. 4, Figure 2 · p. 4

4. Maximum one incoming edge per node

• Admissible graphs have at most n −1 edges for n nodes and are acyclic, with outward edge directions determined by a root node.In a tree, any node can be the root; there are n possible root choices.Source: §4 · p. 5, E. Trees and subtrees · p. 5

E. Trees and subtrees

• Subtrees are classified by node distances from the root, giving a compact description of how branches split and paths reach nodes.For example, (123234) branches at node 2 into (23) and (234).Source: E. Trees and subtrees · p. 5

F. Atomic properties

• Atomic properties can be assigned in infinitely many ways while preserving total energy, so they are not directly useful for training without further constraints.This paper assigns each graph contribution to its root node.Source: F. Atomic properties · p. 5

III. EQUIVARIANCE

• ACE(g) supports translation, rotation, inversion, and permutation symmetry, with rotation and inversion equivariance built using spherical harmonics and generalized Clebsch-Gordan coefficients.The symmetry analysis can transfer to other groups and dimensions when their irreducible representations and generalized Clebsch-Gordan coefficients are available or generatable.Source: §III · pp. 6–7

IV. LOCAL AND GLOBAL ACE

• Graph ACE constructs basis functions for the global property before assigning contributions to atoms, whereas local ACE starts from per-atom environments.Local ACE is the subset of global ACE using star graphs; the two differ from four-body interactions onward for general radial basis functions.Source: §IV · p. 7

A. Completeness and sensitivity

• Tree graphs with different topologies provide distinct whole-system basis functions, and their geometry can make them more sensitive to positional changes than star graphs.The authors argue that narrow star opening angles can make position details numerically challenging to distinguish, while tree edges can have larger angles and comparable lengths.Source: A. Completeness and sensitivity · p. 7
• Tree graphs can extend across several cut-off distances, unlike local ACE star graphs, whose interaction range is limited to twice the cut-off distance.With short pair distances, the star contribution may vanish while a tree contribution remains possible.Source: A. Completeness and sensitivity · p. 8

B. Self-interactions

• Global ACE avoids self-interactions by restricting sums to distinct atoms, and its overcomplete basis can represent corrections even for self-interacting graphs that are not admissible.Some self-interactions may also be beneficial for accuracy and appear in efficient ACE implementations.Source: B. Self-interactions · p. 8

V. QUANTUM MECHANICAL FOUNDATION

• Fourth-moment contributions map to graph terms, but cycle contributions can be less important than pair and three-body terms because interfering angular dependencies can self-cancel.The comparison concerns fourth-moment contributions; onsite matrix elements were ignored in the illustrated analysis.Source: §V · p. 9, Figure 7 · p. 9

A. Tensor decomposition

• Tensor decomposition provides a route to simplifying graph ACE, and the paper initially limits it to chemical indices while retaining the root atom's chemical index for interpretability.A representation removing all chemical indices is also discussed and used in numerical examples.Source: A. Tensor decomposition · p. 10

B. Atomic species

• Chemistry-dependent radial functions and chemical-space basis functions can yield identical representations, making the choice a matter of design.The paper notes that this is also the chemistry-dependent radial-function approach used by MTPs.Source: B. Atomic species · p. 10
• Removing chemical indices from the basis-function index prevents combinatorial growth in expansion coefficients for multicomponent systems.The reduced and full representations are identical with sufficiently many tensor-decomposition terms, while the reduced form enables parameterization of multicomponent systems.Source: B. Atomic species · p. 10

C. Star decomposition and layers

• Stars are indexed by their layer from the root and their number of outgoing edges; a floret is an outgoing edge and its terminal node.The star decomposition also preserves graph symmetry when expansion coefficients are factorized into star coefficients.Source: C. Star decomposition and layers · p. 11

D. Floret picking and subgraph expansion coefficients

• Picking florets in all possible ways generates every subtree and its corresponding expansion coefficient for recursive evaluation.Removing a floret reduces the corresponding star's edge count; a star with zero florets can itself be picked.Source: D. Floret picking and subgraph… · pp. 11–12

E. Recursive evaluation

• Representing subtree coefficients within a tree coefficient makes recursive evaluation possible, illustrated for the (122) graph.The recursive evaluation includes contributions from the tree's (12) and (1) subtrees.Source: E. Recursive evaluation · p. 12

VII. DANDELION APPROXIMATION

• A dandelion is the largest tree in a family of trees obtainable from it by floret picking.The paper focuses on regular dandelions with depth T and P outgoing edges per node, which can accommodate many other trees as subtrees.Source: §VII · p. 12

A. Dandelion expansion coefficients and recursion

• Dandelion contributions and their subtrees are evaluated recursively, initializing with local ACE on the last layer and iterating toward the root.In practice, each layer's local ACE may be stopped at body order K smaller than P; the resulting messages are attached to atoms and transferred to neighbors.Source: A. Dandelion expansion coefficients… · p. 13
• Evaluation of graph ACE and its gradients scales linearly with layer count T and neighbor count N in the dandelion approximation.Each recursion layer extends the effective interaction range by rcut.Source: A. Dandelion expansion coefficients… · p. 13

B. Explicitly including angular contributions

• Angular contributions are indexed by angular momentum l,m, parity p, and radial index n; the first-order term has no angular-momentum coupling.For interatomic potentials, layer 1 uses l=m=0 and even parity, while vectorial or tensorial expansions can use other l and p values.Source: B. Explicitly including angular… · p. 14, §1 · p. 14

1. Independent radial channels

• In the independent-radial-channel variant, Clebsch-Gordan coefficients couple angular character across layers while radial indices remain unmixed.A full tensor decomposition of radial indices is obtained by setting n=n1=n2 and so on.Source: §1 · p. 14, B. Explicitly including angular… · p. 14

2. Coupling radial channels

• Two recursion variants based on different tensor decompositions both allow accurate recursive evaluation of dandelion-approximation graph ACE.Layer-dependent weights can couple radial and chemical indices; the alternatives differ in how channels are handled.Source: §2 · pp. 14–15

A. Multi ACE

• Multi-ACE is contained in graph ACE's design space, and the paper derives its recursive dandelion evaluation from global graph ACE.The paper says multi-ACE cannot be obtained directly from local ACE alone.Source: A. Multi ACE · p. 15

B. Nequip

• NequIP can be understood as a graph ACE realization limited to low-body-order ACE messages, which the paper links to its need for several layers.The paper also frames multi-ACE as a generalization of NequIP to higher-body-order ACE messages.Source: B. Nequip · p. 15

C. Multilayer ACE

• Multilayer ACE is limited to scalar outputs with nonlinear update functions and is a subset of multi-ACE and graph ACE.The paper says it was inspired by electronic-structure relaxation inducing semilocal interactions beyond an atom's immediate neighbor shell.Source: C. Multilayer ACE · p. 15

D. MACE

• MACE is an implementation of multi-ACE whose design choices include distinguishing features from messages and omitting some parity couplings.The authors observe that MACE requires about one order of magnitude more parameters than ACE(g), which they say may be attributed to some design choices.Source: D. MACE · p. 16

E. Allegro

• Allegro’s body order is analogous to its number of layers; the authors speculate that ACE(g) with the same direct neighbors may preserve flexible training functions while accessing non-star graphs.The proposed similarity of mechanisms is presented as speculation.Source: E. Allegro · p. 16

F. Unified atom-centred and message-passing schemes

• Atomic-density correlations resemble graph cluster basis functions, but are not restricted to admissible configurations or acyclic graphs.Source: F. Unified atom-centred and… · p. 16

IX. APPLICATIONS

• The applications assess graph ACE’s accuracy and computational performance on four-atom Si clusters, 10 small organic molecules, flexible 3BPA, and a general-purpose carbon dataset.For Si, the randomized, unrelaxed clusters include 16,180 structures, with 90% used for training.Source: §IX · p. 16, A. Analysis for four-atom Si clusters · p. 16

A. Analysis for four-atom Si clusters

• For four-atom Si clusters, the (123) tree is more accurate than the (111) star, and dandelion models with (122) or (123) largest graphs improve over the explicit (123) graph.At three edges, the (111) star without self-interactions and third-order atomic-base contributions with self-interactions are identical.Source: A. Analysis for four-atom Si clusters · p. 16, Figure 10 · p. 17

approximate product reduction of angular contributions.

• For Si clusters, increasing graph edges further lowers errors, while at six edges ACE(g) significantly improves over ACE(l); two versus three layers is not decisive.The reported error trends are relative; changing hyperparameters can shift errors lower without changing the analysis conclusions.Source: approximate product reduction of… · p. 17

B. Small molecules

• ACE(g) outperforms other models for all 10 MD17 molecules, including when trained on 50 configurations, and for all 3BPA tests.The reported comparisons are in Tables I and II; the 3BPA tests include configurations at elevated temperatures and dihedral slices far from training data.Source: B. Small molecules · p. 17, Table I · p. 18, Table II · p. 18

C. Carbon

• On carbon structures, ACE(g) and MACE improve over ACE(l), especially for clusters; ACE(g) has slightly lower total errors, around 105k parameters, and 200 µs/atom versus MACE’s almost 2.5M parameters and 400 µs/atom.On an 80 Gb GPU, simulations ran up to 12,672 atoms with ACE(g), versus 2,016 with MACE.Source: C. Carbon · p. 18, Figure 11 · p. 19

X. CONCLUSION

• ACE(g) is more accurate than the tested machine-learned interatomic potentials, while also being faster and using an order of magnitude fewer parameters.Tensor decomposition enables recursive evaluation of tree graphs, including their subgraphs, in the dandelion approximation.Source: §X · p. 19

Appendix A: Reducing radial functions

• Radial reduction ties all radial-function indices together, leaving expansion coefficients with a single radial index.The resulting basis functions use the basis of Eq. (49).Source: Appendix A · p. 19

Appendix B: Approximate product reduction of angular contributions

• Approximate product angular reduction assumes a coefficient decomposition whose Clebsch–Gordan factors are independent of k, but may introduce redundant couplings.A more elaborate treatment would decompose the appropriate Clebsch–Gordan coefficients; the product construction is overcomplete.Source: Appendix B · pp. 19–20

a. Basis functions

• The basis separates chemistry from radial functions, with trainable parameters used to differentiate edge chemistry.Source: a. Basis functions · p. 20

b. Radial functions

• Radial functions use either a linear expansion or an MLP; the MLP has three 64-unit hidden layers with SiLu activation and no final-layer activation.The radial bases include simplified spherical Bessel functions and Bessel functions with polynomial envelope, with p = 5 for the latter.Source: b. Radial functions · p. 20

c. Non-linear embedding

• The atomic energy is modeled as a nonlinear function of graph ACE inputs using a single-layer MLP with SiLu activation.The number of units and inputs is varied.Source: c. Non-linear embedding · p. 20

d. Loss functions

• The loss weights energy and force errors across structures and includes L2 regularization of the graph ACE expansion coefficients.For carbon, the energy residual is additionally normalized by the number of atoms in each structure.Source: d. Loss functions · p. 20

e. Model parameters

• Si-cluster models use eight simplified spherical-Bessel radial functions with an 8 Å cutoff, shared across layers, and a linear energy embedding.The settings also use nmax = 8 and lmax = 4 for the atomic base and intermediate couplings.Source: e. Model parameters · p. 20
• Small-molecule models use two layers with variable radial functions, combining a linear first-layer expansion with an MLP expansion on layer zero.The MD17 and 3BPA cutoffs are 4 Å and 4.5 Å, respectively.Source: e. Model parameters · pp. 20–21
• The carbon ACE(g) model uses shared MLP radial functions with a 5 Å cutoff, while the carbon MACE comparison uses a 5 Å radial cutoff.The carbon ACE(g) fit uses Adam followed by L-BFGS-B; the MACE setup uses 256 feature channels.Source: e. Model parameters · p. 21
• The ACE(g) implementation uses tensorpotential with TensorFlow GPU acceleration and double-precision fitting and evaluation.Source: e. Model parameters · p. 21

Appendix D: Comparison of computational performance

• The comparison figure reports ACE(g) and MACE computational performance, but the current ACE(g) setup runs other calculations on one CPU core and transfers data to and from the GPU.The authors state that this implementation could be substantially improved.Source: Figure 12 · p. 21, A comparison of computational… · p. 21

Appendix E: Efficient gradients for graph ACE

• ACE(g) evaluation, including gradients, scales linearly with neighbor count N and layer depth T after changing the summation order.The transformed energy and force evaluation requires effort proportional to TN.Source: Appendix E · pp. 21, 23
• The timing implementation uses TensorFlow automatic differentiation rather than the appendix's explicit gradient formulas.The authors leave comparison of the two approaches' efficiency for future work.Source: Appendix E · p. 21
• Collecting and iterating prefactors avoids the poor scaling of direct iterative summation; the resulting h(t) values must be available on neighboring atoms.The authors note that these values can be communicated to ghost atoms in LAMMPS domain decomposition.Source: Appendix E · pp. 22–23

• The algorithm assumes directed bonds have lookup-accessible inverse bonds and that tensors can be attached to atoms and communicated to ghost atoms.If inverse-bond indices are unavailable, values for the inverse bonds must also be computed and stored.Source: §1 · p. 23
• Recursive evaluation of local ACE is performed on each layer and atom before the gradient algorithm is applied.Source: §1 · p. 23
• The algorithm initializes and communicates layerwise quantities, iterates over layers and neighboring atoms, and accumulates forces.Source: §1 · p. 23
• During each layer update, the procedure loops over neighbors to form atomic-base and prefactor quantities, then communicates results to ghost atoms.Source: §1 · p. 23

• The basis functions combine radial functions with spherical harmonics, with parity determined by angular index l.Source: §2 · p. 23
• Effective radial functions are computed on each layer and used to obtain the atomic base.Source: §2 · pp. 23–24
• Each layer applies a local ACE expansion, and the final layer produces the graph ACE representation.Source: §2 · pp. 23–24
• The energy is computed from the resulting graph ACE representation.Source: §2 · p. 24