September 28, 2025Open Access

Constructing Invariant and Equivariant Operations by Symmetric Tensor Network

Key Points

Invariant and equivariant operations are crucial for enhancing neural networks' capability to handle geometric data, and this method effectively constructs them.
Central to the approach is the use of symmetric tensor networks, which provide a clear graphical representation for deriving these operations.
Our method accommodates inputs of various ranks, including Cartesian and spherical tensors, facilitating broader application in machine learning.
This framework supports the design of interaction messages in geometry graph neural networks and models for material behavior.

Abstract

Design of neural networks that incorporate symmetry is crucial for geometric deep learning. Central to this effort is the development of invariant and equivariant operations. This works presents a systematic method for constructing valid invariant and equivariant operations. It can handle inputs and outputs in the form of Cartesian tensors with different rank, as well as spherical tensors with different types. In addition, our method features a graphical representation utilizing the symmetric tensor network, which simplifies both the proofs and constructions related to invariant and equivariant functions. We also apply this approach to design the equivariant interaction message for the geometry graph neural network, and equivariant machine learning model to learn the constitutive law of materials.

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