# Hierarchical Learning in Euclidean Neural Networks

arXiv: https://arxiv.org/abs/2210.04766 · HTML: https://mlsanity.com/paper/2210.04766/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 18 references that are in the map.
- First posted on arXiv 10 Oct 2022.

## Builds on (most cited first, 8 of 8)

- [E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentials](https://mlsanity.com/paper/2101.03164/) (2021)
- [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)
- [Geometric and Physical Quantities Improve E(3) Equivariant Message Passing](https://mlsanity.com/paper/2110.02905/) (2021)
- [e3nn: Euclidean Neural Networks](https://mlsanity.com/paper/2207.09453/) (2022)
- [Scalars are universal: Equivariant machine learning, structured like classical physics](https://mlsanity.com/paper/2106.06610/) (2021)
- [The Design Space of E(3)-Equivariant Atom-Centered Interatomic Potentials](https://mlsanity.com/paper/2205.06643/) (2022)
- [Relevance of Rotationally Equivariant Convolutions for Predicting Molecular Properties](https://mlsanity.com/paper/2008.08461/) (2020)

## Abstract (arXiv)

Equivariant machine learning methods have shown wide success at 3D learning applications in recent years. These models explicitly build in the reflection, translation and rotation symmetries of Euclidean space and have facilitated large advances in accuracy and data efficiency for a range of applications in the physical sciences. An outstanding question for equivariant models is why they achieve such larger-than-expected advances in these applications. To probe this question, we examine the role of higher order (non-scalar) features in Euclidean Neural Networks (texttte3nn). We focus on the previously studied application of texttte3nn to the problem of electron density prediction, which allows for a variety of non-scalar outputs, and examine whether the nature of the output (scalar l=0, vector l=1, or higher order l>1) is relevant to the effectiveness of non-scalar hidden features in the network. Further, we examine the behavior of non-scalar features throughout training, finding a natural hierarchy of features by l, reminiscent of a multipole expansion. We aim for our work to ultimately inform design principles and choices of domain applications for tt e3nn networks.