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TensorNet: Cartesian Tensor Representations for Efficient Learning of Molecular Potentials

arXiv 2306.06482 ↗First posted 10 Jun 2023cs.LG physics.chem-ph physics.comp-phcited by 53

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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.

Key facts. Cited by 53 papers in the map (data through 7 Oct 2026). Builds on 29 of its 45 references that are in the map. First posted on arXiv 10 Jun 2023.

Builds on

29 of 45 references.

References · 29

  1. 2019PyTorch: An Imperative Style, High-Performance Deep Learning Librarycited by 20,160
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  8. 2021E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentialscited by 701
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17 more
  1. 20183D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Datacited by 348
  2. 2021GemNet: Universal Directional Graph Neural Networks for Moleculescited by 342
  3. 2019Cormorant: Covariant Molecular Neural Networkscited by 262
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  5. 2022Learning Local Equivariant Representations for Large-Scale Atomistic Dynamicscited by 254
  6. 2021Geometric and Physical Quantities Improve E(3) Equivariant Message Passingcited by 225
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Cited by year (2026 to 7 Oct)

Citing papers per year: 2023: 2; 2024: 23; 2025: 14; 2026: 14.

53 papers.

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  1. 2025Position: Graph Learning Will Lose Relevance Due To Poor Benchmarkscited by 52
  2. 2023OpenMM 8: Molecular Dynamics Simulation with Machine Learning Potentialscited by 44
  3. 2024ET-Flow: Equivariant Flow-Matching for Molecular Conformer Generationcited by 41
  4. 2024Enabling Efficient Equivariant Operations in the Fourier Basis via Gaunt Tensor Productscited by 35
  5. 2024Cartesian atomic cluster expansion for machine learning interatomic potentialscited by 31
  6. 2024TorchMD-Net 2.0: Fast Neural Network Potentials for Molecular Simulationscited by 15
  7. 2024A predictive machine learning force field framework for liquid electrolyte developmentcited by 13
  8. 2024Higher-Rank Irreducible Cartesian Tensors for Equivariant Message Passingcited by 13
  9. 2024Beyond Canonicalization: How Tensorial Messages Improve Equivariant Message Passingcited by 13
  10. 2024Probing the effects of broken symmetries in machine learningcited by 13
  11. 2025Enhancing the Scalability and Applicability of Kohn-Sham Hamiltonians for Molecular Systemscited by 13
  12. 2025E2Former: An Efficient and Equivariant Transformer with Linear-Scaling Tensor Productscited by 12
38 more
  1. 2026AIRA_2: Overcoming Bottlenecks in AI Research Agentscited by 10
  2. 2025ELECTRA: A Cartesian Network for 3D Charge Density Prediction with Floating Orbitalscited by 9
  3. 2024Nutmeg and SPICE: Models and Data for Biomolecular Machine Learningcited by 7
  4. 2024Learning local equivariant representations for quantum operatorscited by 7
  5. 2024Generating Highly Designable Proteins with Geometric Algebra Flow Matchingcited by 7
  6. 2023Synthetic pre-training for neural-network interatomic potentialscited by 6
  7. 2024High-Rank Irreducible Cartesian Tensor Decomposition and Bases of Equivariant Spacescited by 6
  8. 2025Efficient Prediction of SO(3)-Equivariant Hamiltonian Matrices via SO(2) Local Framescited by 5
  9. 2025Spectral/Spatial Tensor Atomic Cluster Expansion with Universal Embeddings in Cartesian Spacecited by 5
  10. 2025DistMLIP: A Distributed Inference Platform for Machine Learning Interatomic Potentialscited by 4
  11. 2024Broadening the Scope of Neural Network Potentials through Direct Inclusion of Additional Molecular Attributescited by 3
  12. 2024Grappa -- A Machine Learned Molecular Mechanics Force Fieldcited by 3
  13. 2024Tensor learning with orthogonal, Lorentz, and symplectic symmetriescited by 3
  14. 2024CodeRefine: A Pipeline for Enhancing LLM-Generated Code Implementations of Research Paperscited by 3
  15. 2024On the design space between molecular mechanics and machine learning force fieldscited by 3
  16. 2024AMARO: All Heavy-Atom Transferable Neural Network Potentials of Protein Thermodynamicscited by 3
  17. 2025CLOUD: A Scalable and Physics-Informed Foundation Model for Crystal Representation Learningcited by 3
  18. 2026Elign: Equivariant Diffusion Model Alignment from Foundational Machine Learning Force Fieldscited by 3
  19. 2025Data Fusion of Deep Learned Molecular Embeddings for Property Predictioncited by 2
  20. 2025A Cartesian-3j Framework for Machine Learning Interatomic Potentialscited by 2
  21. 2026Smooth Dynamic Cutoffs for Machine Learning Interatomic Potentialscited by 2
  22. 2024FreeCG: Free the Design Space of Clebsch-Gordan Transform for Machine Learning Force Fieldscited by 1
  23. 2025QuantumBind-RBFE: Accurate Relative Binding Free Energy Calculations Using Neural Network Potentialscited by 1
  24. 2025Optimal Invariant Bases for Atomistic Machine Learningcited by 1
  25. 2025Hierarchical geometric deep learning enables scalable analysis of molecular dynamicscited by 1
  26. 2026Geometric Algebra Meets Cartesian Tensors: Higher-Order Equivariance for Interatomic Potentialscited by 1
  27. 2024Ensemble Learning of Machine Learning Force Fieldscited by 0
  28. 2024PlayMolecule pKAce: Small Molecule Protonation through Equivariant Neural Networkscited by 0
  29. 2024On Machine Learning Approaches for Protein-Ligand Binding Affinity Predictioncited by 0
  30. 2024OpenQDC: Open Quantum Data Commonscited by 0
  31. 2025Stable and Accurate Orbital-Free DFT Powered by Machine Learningcited by 0
  32. 2026AceFF: A State-of-the-Art Machine Learning Potential for Small Moleculescited by 0
  33. 2026Efficient Equivariant High-Order Crystal Tensor Prediction via Cartesian Local-Environment Many-Body Couplingcited by 0
  34. 2026AQVolt26: High-Temperature r2SCAN Halide Dataset for Universal ML Potentials and Solid-State Batteriescited by 0
  35. 2026Diagnosing Spectral Ceilings in Equivariant Neural Force Fieldscited by 0
  36. 2026From Single-Step Edit Response to Multi-Step Molecular Optimizationcited by 0
  37. 2026Tensor Channel Equivariant Graph Neural Networks for Molecular Polarizability Predictioncited by 0
  38. 2026Equivariant Graph Neural Networks Improve Optical Spectra Prediction for Materials Screeningcited by 0

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