SPACE reaches arbitrarily high body order with compact equivariant tensor products, dropping the low-order cluster-expansion truncation

Introduces SPACE (Smooth Physical Architecture with Compact Equivariants), which incorporates locality, smoothness and symmetry priors into short-range discretizations of the descriptors while letting body order grow to the typical number of atoms within the model's receptive field; despite similarities to ACE-inspired models, it is not a cluster expansion because it effectively incorporates all orders of interactions. Many equivariant symmetric architectures historically assume the interatomic potential is well approximated by a convergent cluster expansion, i.e. a decomposition into pairs, triplets, quadruplets and higher-order tuples; SPACE does not assume low-order truncation of the body-ordered series. The claim rests on architectural construction and theoretical derivation: the main text formalizes equivariant tensor products, the compact representation and message-passing modules, and Appendix B gives full architecture details; no regression benchmark numbers are reported.

Constructs a physically inspired radial basis by multiplying the numerator of the Rayleigh quotient by an exponential of the distance, penalizing non-smooth functions exponentially more as atoms separate, reflecting the exponential decay of atomic orbitals; the basis generalizes to high-body-order descriptors and its Rayleigh quotient values can serve as diagonal regularization entries. Earlier constructions based on spherical Bessel functions impose the same level of smoothness within the whole atom-centered sphere, which the authors argue is not a faithful physical prior; the new construction brings a distance-dependent decay law into short-range descriptions. Derived in Sections II.3 and II.4 and Appendix A: integration by parts yields a linear operator, the radial equation is solved numerically by expanding on trigonometric functions, and solutions are observed to vanish almost completely well before the cutoff; radial basis functions are illustrated for selected parameters.

Uses tensor products in the compact equivariant representation, where they reduce to matrix multiplication between square matrices, scaling as rather than for naive tensor products in the spherical representation and equivalent to the full tensor product truncated at a maximum degree; this supports six or more tensor products per GNN layer, with six consecutive products reaching a correlation order of and at least over multiple message-passing layers, exceeding typical neighbor counts. Common equivariant architectures typically use two tensor products per layer; SPACE exploits the efficiency of the compact representation to raise the number of tensor products substantially, reaching very high correlation orders without a dramatic increase in computational effort. The tensor-product form and scaling come from the cited compact-covariants work; Sections II.5 and II.6 and Appendix B.2 describe the implementation path, with features transformed between coupled and uncoupled representations and self-products executed as matrix multiplication without further CG coefficient matrices inside the GNN backbone.

The architecture is SO(3)-equivariant rather than O(3)-equivariant, applying inversion augmentation at training time and recovering inversion symmetry exactly at prediction time at the cost of a single additional evaluation, which the authors describe as entirely harmless. Naive O(3)-equivariant networks must carry separate even and odd representations, making tensor products about four times more expensive than SO(3) tensor products, so in practice they often simply drop all odd representations, potentially at the expense of completeness and representation power; SPACE instead does not enforce inversion symmetry and recovers it at prediction time. The design is described in Section II.6 with an architecture illustration and detailed description in Appendix B; no quantitative comparison experiments are reported.

The work targets architecture design for machine-learning interatomic potentials in atomistic modeling, for readers who want equivariant models without assuming low-order truncation of the cluster expansion. Methodologically, the physically inspired radial basis generalizes to high-body-order descriptors of any order, the compact-representation tensor products suit modern machine-learning frameworks and GPU execution, and the SO(3)-equivariant design with training-time inversion augmentation and exact recovery at prediction time via one extra evaluation can be reused in other equivariant networks. Hyperparameters are chosen to cover the maximum angular order of the target, for instance at least the corresponding threshold when learning dipoles, a vectorial target with maximum angular order 1. The authors explicitly frame SPACE as a demonstration that the design space of equivariant architectures still contains poorly explored regions that may offer a better balance between expressivity and computational effort.

The authors note that expressivity and physical preconditioning of the radial basis do not automatically translate into regression performance, and that careful testing across different types and sizes of datasets is needed to assess the concrete impact of these architectural choices, so SPACE is currently presented as a demonstration of design-space exploration rather than a performance conclusion. A careful reader might also watch how the basis truncation threshold and lengthscale parameter affect the accuracy-efficiency trade-off; the choice between sequences of compact equivariant products and equivariant nonlinear layers (the authors report numerical instabilities when experimenting with an equivariant hyperbolic tangent nonlinear layer, and found it less flexible than a sequence of compact equivariant products); and the practical cost of recovering inversion symmetry at prediction time. In addition, although this is a full-text parse, formulas and figures appear as placeholders in the loaded text, so details tied to specific numerical values cannot be verified from it.