# TensorNet: Cartesian Tensor Representations for Efficient Learning of Molecular Potentials

arXiv: https://arxiv.org/abs/2306.06482 · HTML: https://mlsanity.com/paper/2306.06482/

## Key facts

- Cited by 54 papers in the map (data through 14 Oct 2026).
- Builds on 29 of its 45 references that are in the map.
- First posted on arXiv 10 Jun 2023.

## Builds on (most cited first, 29 of 29)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [Graph Neural Networks: A Review of Methods and Applications](https://mlsanity.com/paper/1812.08434/) (2018)
- [Relational inductive biases, deep learning, and graph networks](https://mlsanity.com/paper/1806.01261/) (2018)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentials](https://mlsanity.com/paper/2101.03164/) (2021)
- [SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networks](https://mlsanity.com/paper/2006.10503/) (2020)
- [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)
- [Equivariant message passing for the prediction of tensorial properties and molecular spectra](https://mlsanity.com/paper/2102.03150/) (2021)
- [3D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Data](https://mlsanity.com/paper/1807.02547/) (2018)
- [GemNet: Universal Directional Graph Neural Networks for Molecules](https://mlsanity.com/paper/2106.08903/) (2021)
- [Cormorant: Covariant Molecular Neural Networks](https://mlsanity.com/paper/1906.04015/) (2019)
- [Learning Local Equivariant Representations for Large-Scale Atomistic Dynamics](https://mlsanity.com/paper/2204.05249/) (2022)
- [Fast and Uncertainty-Aware Directional Message Passing for Non-Equilibrium Molecules](https://mlsanity.com/paper/2011.14115/) (2020)
- [Geometric and Physical Quantities Improve E(3) Equivariant Message Passing](https://mlsanity.com/paper/2110.02905/) (2021)
- [TorchMD-NET: Equivariant Transformers for Neural Network based Molecular Potentials](https://mlsanity.com/paper/2202.02541/) (2022)
- [Machine learning for molecular simulation](https://mlsanity.com/paper/1911.02792/) (2019)
- [The Design Space of E(3)-Equivariant Atom-Centered Interatomic Potentials](https://mlsanity.com/paper/2205.06643/) (2022)
- [Less is more: sampling chemical space with active learning](https://mlsanity.com/paper/1801.09319/) (2018)
- [SPICE, A Dataset of Drug-like Molecules and Peptides for Training Machine Learning Potentials](https://mlsanity.com/paper/2209.10702/) (2022)
- [Scaling the leading accuracy of deep equivariant models to biomolecular simulations of realistic size](https://mlsanity.com/paper/2304.10061/) (2023)
- [SchNetPack 2.0: A neural network toolbox for atomistic machine learning](https://mlsanity.com/paper/2212.05517/) (2022)
- [Evaluation of the MACE Force Field Architecture: from Medicinal Chemistry to Materials Science](https://mlsanity.com/paper/2305.14247/) (2023)
- [Machine learning of solvent effects on molecular spectra and reactions](https://mlsanity.com/paper/2010.14942/) (2020)
- [Spatial Attention Kinetic Networks with E(n)-Equivariance](https://mlsanity.com/paper/2301.08893/) (2023)
- [Molecular Dynamics with Neural-Network Potentials](https://mlsanity.com/paper/1812.07676/) (2018)

## Built on by (most cited first, 30 of 54)

- [Position: Graph Learning Will Lose Relevance Due To Poor Benchmarks](https://mlsanity.com/paper/2502.14546/) (2025)
- [OpenMM 8: Molecular Dynamics Simulation with Machine Learning Potentials](https://mlsanity.com/paper/2310.03121/) (2023)
- [ET-Flow: Equivariant Flow-Matching for Molecular Conformer Generation](https://mlsanity.com/paper/2410.22388/) (2024)
- [Enabling Efficient Equivariant Operations in the Fourier Basis via Gaunt Tensor Products](https://mlsanity.com/paper/2401.10216/) (2024)
- [Cartesian atomic cluster expansion for machine learning interatomic potentials](https://mlsanity.com/paper/2402.07472/) (2024)
- [TorchMD-Net 2.0: Fast Neural Network Potentials for Molecular Simulations](https://mlsanity.com/paper/2402.17660/) (2024)
- [Beyond Canonicalization: How Tensorial Messages Improve Equivariant Message Passing](https://mlsanity.com/paper/2405.15389/) (2024)
- [Enhancing the Scalability and Applicability of Kohn-Sham Hamiltonians for Molecular Systems](https://mlsanity.com/paper/2502.19227/) (2025)
- [A predictive machine learning force field framework for liquid electrolyte development](https://mlsanity.com/paper/2404.07181/) (2024)
- [Higher-Rank Irreducible Cartesian Tensors for Equivariant Message Passing](https://mlsanity.com/paper/2405.14253/) (2024)
- [Probing the effects of broken symmetries in machine learning](https://mlsanity.com/paper/2406.17747/) (2024)
- [E2Former: An Efficient and Equivariant Transformer with Linear-Scaling Tensor Products](https://mlsanity.com/paper/2501.19216/) (2025)
- [AIRA_2: Overcoming Bottlenecks in AI Research Agents](https://mlsanity.com/paper/2603.26499/) (2026)
- [ELECTRA: A Cartesian Network for 3D Charge Density Prediction with Floating Orbitals](https://mlsanity.com/paper/2503.08305/) (2025)
- [Learning local equivariant representations for quantum operators](https://mlsanity.com/paper/2407.06053/) (2024)
- [Nutmeg and SPICE: Models and Data for Biomolecular Machine Learning](https://mlsanity.com/paper/2406.13112/) (2024)
- [Generating Highly Designable Proteins with Geometric Algebra Flow Matching](https://mlsanity.com/paper/2411.05238/) (2024)
- [Synthetic pre-training for neural-network interatomic potentials](https://mlsanity.com/paper/2307.15714/) (2023)
- [High-Rank Irreducible Cartesian Tensor Decomposition and Bases of Equivariant Spaces](https://mlsanity.com/paper/2412.18263/) (2024)
- [Efficient Prediction of SO(3)-Equivariant Hamiltonian Matrices via SO(2) Local Frames](https://mlsanity.com/paper/2506.09398/) (2025)
- [Spectral/Spatial Tensor Atomic Cluster Expansion with Universal Embeddings in Cartesian Space](https://mlsanity.com/paper/2509.14961/) (2025)
- [DistMLIP: A Distributed Inference Platform for Machine Learning Interatomic Potentials](https://mlsanity.com/paper/2506.02023/) (2025)
- [On the design space between molecular mechanics and machine learning force fields](https://mlsanity.com/paper/2409.01931/) (2024)
- [Broadening the Scope of Neural Network Potentials through Direct Inclusion of Additional Molecular Attributes](https://mlsanity.com/paper/2403.15073/) (2024)
- [Grappa -- A Machine Learned Molecular Mechanics Force Field](https://mlsanity.com/paper/2404.00050/) (2024)
- [Tensor learning with orthogonal, Lorentz, and symplectic symmetries](https://mlsanity.com/paper/2406.01552/) (2024)
- [CodeRefine: A Pipeline for Enhancing LLM-Generated Code Implementations of Research Papers](https://mlsanity.com/paper/2408.13366/) (2024)
- [AMARO: All Heavy-Atom Transferable Neural Network Potentials of Protein Thermodynamics](https://mlsanity.com/paper/2409.17852/) (2024)
- [CLOUD: A Scalable and Physics-Informed Foundation Model for Crystal Representation Learning](https://mlsanity.com/paper/2506.17345/) (2025)
- [A Cartesian-3j Framework for Machine Learning Interatomic Potentials](https://mlsanity.com/paper/2512.16882/) (2025)

## Abstract (arXiv)

The development of efficient machine learning models for molecular systems representation is becoming crucial in scientific research. We introduce TensorNet, an innovative O(3)-equivariant message-passing neural network architecture that leverages Cartesian tensor representations. By using Cartesian tensor atomic embeddings, feature mixing is simplified through matrix product operations. Furthermore, the cost-effective decomposition of these tensors into rotation group irreducible representations allows for the separate processing of scalars, vectors, and tensors when necessary. Compared to higher-rank spherical tensor models, TensorNet demonstrates state-of-the-art performance with significantly fewer parameters. For small molecule potential energies, this can be achieved even with a single interaction layer. As a result of all these properties, the model's computational cost is substantially decreased. Moreover, the accurate prediction of vector and tensor molecular quantities on top of potential energies and forces is possible. In summary, TensorNet's framework opens up a new space for the design of state-of-the-art equivariant models.