Equivariant Interatomic Potentials without Tensor Products

TL;DR

Geodite addresses the trade-off between expressiveness and speed in equivariant interatomic potentials by removing Clebsch-Gordan tensor products and instead using inner-product based equivariant interactions with physically motivated priors. The model, Geodite-MP, is trained on the Materials Project trajectory dataset (MPtrj) and evaluated on Matbench Discovery, MDR, diatomic binding curves, and SSE MD, achieving competitive accuracy while running $3$–$5 imes$ faster than similar tensor-product-based models. The design includes a residual energy formulation with vacuum embeddings, smoothness constraints to avoid discontinuities, explicit short-range repulsion with learnable screening, and per-edge density normalization to stabilize training. These results demonstrate robust PES accuracy, smooth dynamics, and substantial efficiency gains, enabling large-scale simulations and high-throughput screening in inorganic materials.

Abstract

Foundational machine-learned interatomic potentials have emerged as powerful tools for atomistic simulations, promising near first-principles accuracy across diverse chemical spaces at a fraction of the cost of quantum-mechanical calculations. However, the most accurate equivariant architectures rely on Clebsch-Gordan tensor products whose computational cost scales steeply with angular resolution, creating a trade-off between model expressiveness and inference speed that ultimately limits practical applications. Here we introduce Geodite, an equivariant message-passing architecture that replaces tensor products while incorporating physical priors to ensure smooth, well-behaved potential energy surfaces. Trained on the Materials Project trajectories dataset of inorganic crystals, Geodite-MP achieves accuracy competitive with leading methods on benchmarks for materials stability prediction, thermal conductivity, phonon-derived properties, and nanosecond-scale molecular dynamics, while running $3\text{--}5\times$ faster than models performing similarly. By combining predictive accuracy, computational efficiency, and physicality, Geodite enables faster large-scale atomistic simulations and high-throughput screening that would otherwise be computationally prohibitive.

Figures (5)

• Figure 1: Overview of the Geodite architecture.(A) The system's potential energy $E$, atomic forces $\mathbf{F}$, and stresses $\bm{\sigma}$ are predicted by an equivariant message-passing neural network based on atomic positions $\mathbf{r}$ and types $\mathbf{Z}$. (B--C) Initial scalar node and edge features, $\textbf{h}^0_i$ and $\textbf{t}_{ij}^0$, are created based on atomic types and system's topology. (D) In an interaction layer, scalar and steerable node representations, $\textbf{h}^\mathrm{k}_i$ and $\tilde{\textbf{h}}^\mathrm{k}_i$, are refined based on neighbor information using modified self-attention and spatial filtering. (E) In the equivariant coupling block, information is exchanged between invariant and equivariant features to produce the output embeddings, $\textbf{h}^\mathrm{k+1}_i$ and $\tilde{\textbf{h}}^\mathrm{k+1}_i$, of the respective message passing layer. (F) Similarly, in the edge update, the revised steerable features and spherical harmonics representation of edge vectors, $\tilde{\textbf{r}}_{ij}$, are used to enhance the learnable edge representation, $\textbf{t}_{ij}^{k+1}$. $\oplus$ denotes aggregation over neighbors, $\cdot\|\cdot$ denotes concatenation, $\bigcirc$ denotes element-wise product, $\|\cdot\|$ denotes vector norm, and $\phi_{\mathrm{eff}}$ and $\rho$ denote the effective radial basis embedding and learned atomic density factor, respectively.
• Figure 2: Computational efficiency and Matbench Discovery accuracy trade-off across .A. Inference time per molecular dynamics step as a function of system size. All models show increasing computational cost with system size, with different scaling behaviors emerging beyond 500 atoms. Beyond 500 atoms, all models scale linearly on a log-log plot, with vertical shifts reflecting differences in computational cost per atom. B. Relationship between Matbench Discovery performance metrics (F1 score and $\kappa_{\textrm{SRME}}$) and computational efficiency. Marker size indicates inference speed, with smaller markers representing faster models. Models occupy different positions along the accuracy-efficiency trade-off, with some prioritizing predictive performance and others computational throughput.
• Figure 3: Binding curves of homonuclear diatomics and asymptotic behavior.A. Binding energy (top) and force (bottom) curves for homonuclear diatomics of H, P, and Cu as predicted by Geodite-MP and other . B. Layer-wise decomposition of the binding energy surface for Geodite-MP demonstrates that the attenuation function (top) suppresses neural network contributions at short range, which allows the ZBL repulsive term to dominate as atoms approach unphysical overlap. Individual layer contributions decrease as the interatomic distance falls below the sum of covalent radii, while the repulsion term increases monotonically.
• Figure 4: Performance of on solid-state electrolyte molecular dynamics.A. Benchmark scores aggregated across 49 solid-state electrolyte trajectories at various temperatures. Each dot represents a single system colored by the mobile ion species (Li, Cs, Cu, Na, or O). Scores quantify agreement between MLIP and , with values closer to 1.0 indicating better reproduction. Models are ranked by mean score shown in parentheses. B. Representative comparing MLIP predictions (colored lines) to reference (black dashed) for three systems: CuI at 700 K (top), NaBH$_4$ at 300 K (middle), and Li$_7$P$_3$S$_{11}$ at 600 K (bottom). Insets show the crystal structure with mobile ions highlighted. Scores indicate quantitative agreement for the specific element pair shown.
• Figure S1: Diatomics binding curves across elements in MPtrj.