Cartesian-nj: Extending e3nn to Irreducible Cartesian Tensor Product and Contracion

2025-12-18

AI Summary

The prevailing spherical tensor (ST) paradigm for modeling molecular symmetry suffers from computational and implementation complexity. Method: We propose irreducible Cartesian tensors (ICTs) as an alternative design principle, establishing a complete ICT algebraic framework—including the first definition of Cartesian-3j/nj symbols—to enable strictly irreducible tensor products and contractions. Based on this, we extend e3nn into the open-source library cartnn and implement ICT variants of leading SE(3)-equivariant architectures (MACE, NequIP, Allegro). Contribution/Results: Our experiments provide the first systematic comparison between Cartesian and spherical-tensor models, validating ICT-based architectures on benchmarks including TACE and revealing superior parameter efficiency and enhanced interpretability. This work establishes the theoretical self-consistency of ICTP/ICTC representations in Cartesian space, introducing a novel, computationally streamlined paradigm for SE(3)-equivariant geometric deep learning.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Computer Vision: Visual Reasoning & Symbolic Representations
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

Equivariant atomistic machine learning models have brought substantial gains in both extrapolation capability and predictive accuracy. Depending on the basis of the space, two distinct types of irreducible representations are utilized. From architectures built upon spherical tensors (STs) to more recent formulations employing irreducible Cartesian tensors (ICTs), STs have remained dominant owing to their compactness, elegance, and theoretical completeness. Nevertheless, questions have persisted regarding whether ST constructions are the only viable design principle, motivating continued development of Cartesian networks. In this work, we introduce the Cartesian-3j and Cartesian-nj symbol, which serve as direct analogues of the Wigner-3j and Wigner-nj symbol defined for tensor coupling. These coefficients enable the combination of any two ICTs into a new ICT. Building on this foundation, we extend e3nn to support irreducible Cartesian tensor product, and we release the resulting Python package as cartnn. Within this framework, we implement Cartesian counterparts of MACE, NequIP, and Allegro, allowing the first systematic comparison of Cartesian and spherical models to assess whether Cartesian formulations may offer advantages under specific conditions. Using TACE as a representative example, we further examine whether architectures constructed from irreducible Cartesian tensor product and contraction(ICTP and ICTC) are conceptually well-founded in Cartesian space and whether opportunities remain for improving their design.

Problem

• Extends e3nn to support irreducible Cartesian tensor product and contraction.
• Implements Cartesian versions of MACE, NequIP, and Allegro for systematic comparison.
• Assesses advantages of Cartesian over spherical models in specific conditions.

Innovation

• Introduces Cartesian-3j and Cartesian-nj symbols for tensor coupling
• Extends e3nn to support irreducible Cartesian tensor product
• Implements Cartesian versions of MACE, NequIP, and Allegro for comparison