# Spectral/Spatial Tensor Atomic Cluster Expansion with Universal Embeddings in Cartesian Space

arXiv: https://arxiv.org/abs/2509.14961 · HTML: https://mlsanity.com/paper/2509.14961/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 76 references that are in the map.
- First posted on arXiv 18 Sep 2025.

## Builds on (most cited first, 22 of 22)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentials](https://mlsanity.com/paper/2101.03164/) (2021)
- [SchNet: A continuous-filter convolutional neural network for modeling quantum interactions](https://mlsanity.com/paper/1706.08566/) (2017)
- [Directional Message Passing for Molecular Graphs](https://mlsanity.com/paper/2003.03123/) (2020)
- [MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields](https://mlsanity.com/paper/2206.07697/) (2022)
- [Learning Local Equivariant Representations for Large-Scale Atomistic Dynamics](https://mlsanity.com/paper/2204.05249/) (2022)
- [Reducing SO(3) Convolutions to SO(2) for Efficient Equivariant GNNs](https://mlsanity.com/paper/2302.03655/) (2023)
- [The Design Space of E(3)-Equivariant Atom-Centered Interatomic Potentials](https://mlsanity.com/paper/2205.06643/) (2022)
- [Learning Smooth and Expressive Interatomic Potentials for Physical Property Prediction](https://mlsanity.com/paper/2502.12147/) (2025)
- [DeePMD-kit: A deep learning package for many-body potential energy representation and molecular dynamics](https://mlsanity.com/paper/1712.03641/) (2017)
- [TensorNet: Cartesian Tensor Representations for Efficient Learning of Molecular Potentials](https://mlsanity.com/paper/2306.06482/) (2023)
- [Cartesian atomic cluster expansion for machine learning interatomic potentials](https://mlsanity.com/paper/2402.07472/) (2024)
- [Latent Ewald summation for machine learning of long-range interactions](https://mlsanity.com/paper/2408.15165/) (2024)
- [Evaluation of the MACE Force Field Architecture: from Medicinal Chemistry to Materials Science](https://mlsanity.com/paper/2305.14247/) (2023)
- [The dark side of the forces: assessing non-conservative force models for atomistic machine learning](https://mlsanity.com/paper/2412.11569/) (2024)
- [High-performance training and inference for deep equivariant interatomic potentials](https://mlsanity.com/paper/2504.16068/) (2025)
- [Higher-Rank Irreducible Cartesian Tensors for Equivariant Message Passing](https://mlsanity.com/paper/2405.14253/) (2024)
- [A universal augmentation framework for long-range electrostatics in machine learning interatomic potentials](https://mlsanity.com/paper/2507.14302/) (2025)
- [QeMFi: A Multifidelity Dataset of Quantum Chemical Properties of Diverse Molecules](https://mlsanity.com/paper/2406.14149/) (2024)
- [Learning charges and long-range interactions from energies and forces](https://mlsanity.com/paper/2412.15455/) (2024)
- [High-Rank Irreducible Cartesian Tensor Decomposition and Bases of Equivariant Spaces](https://mlsanity.com/paper/2412.18263/) (2024)
- [Machine learning interatomic potential can infer electrical response](https://mlsanity.com/paper/2504.05169/) (2025)

## Built on by (most cited first, 6 of 6)

- [A Cartesian-3j Framework for Machine Learning Interatomic Potentials](https://mlsanity.com/paper/2512.16882/) (2025)
- [Edge Cluster Expansion with Radial Rotary Attention for Interatomic Potentials](https://mlsanity.com/paper/2607.10664/) (2026)
- [Neural-Network Solutions to Real-Space Charge Density and Generalization](https://mlsanity.com/paper/2609.14906/) (2026)
- [A strategic roadmap for an atomistic machine-learning ecosystem](https://mlsanity.com/paper/2609.39090/) (2026)
- [Origins of Universal Machine Learning Force-Field Errors in Multicomponent Materials](https://mlsanity.com/paper/2610.09837/) (2026)
- [LunaTP: Enabling Efficient and Scalable Tensor Product for Equivariant MLIPs](https://mlsanity.com/paper/2610.18393/) (2026)

## Abstract (arXiv)

Equivariant atomistic machine learning models have largely been built on spherical-tensor representations, where explicit angular-momentum coupling introduces substantial complexity and systematic extensions beyond energies and forces remain challenging, often requires problem-specific architectural choices. Here we introduce the Tensor Atomic Cluster Expansion (TACE), which unifies scalar and tensorial modeling in Cartesian and space by decomposing local environments into irreducible Cartesian tensors (ICT) constructing a controlled many-body hierarchy with atomic cluster expansion (ACE). In addition to performing ACE in the frequency domain, we propose an efficient Clebsch-Gordan-free alternative in the spatial domain. TACE provides universal invariant (e.g., fidelity tags and charges) and equivariant (e.g., external electric fields and non-collinear magnetic moments) embeddings and predicted tensorial observables are handled on equal footing and enabling explicit control at inference. We demonstrate the accuracy, stability, and efficiency across finite molecules and extended materials, including in-domain and out-of-domain benchmarks, spectra, Hessian, external-field responses, charged systems, and multi-fidelity/head training. We further show its robustness on nonequilibrium/reactive datasets and controlled scaling when extending to large foundation model datasets.