# Towards Harmonization of SO(3)-Equivariance and Expressiveness: a Hybrid Deep Learning Framework for Electronic-Structure Hamiltonian Prediction

arXiv: https://arxiv.org/abs/2401.00744 · HTML: https://mlsanity.com/paper/2401.00744/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 32 references that are in the map.
- First posted on arXiv 1 Jan 2024.

## Builds on (most cited first, 16 of 16)

- [Spatial Transformer Networks](https://mlsanity.com/paper/1506.02025/) (2015)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networks](https://mlsanity.com/paper/2006.10503/) (2020)
- [Steerable CNNs](https://mlsanity.com/paper/1612.08498/) (2016)
- [Equiformer: Equivariant Graph Attention Transformer for 3D Atomistic Graphs](https://mlsanity.com/paper/2206.11990/) (2022)
- [EquiformerV2: Improved Equivariant Transformer for Scaling to Higher-Degree Representations](https://mlsanity.com/paper/2306.12059/) (2023)
- [Equivariance Through Parameter-Sharing](https://mlsanity.com/paper/1702.08389/) (2017)
- [Artificial Intelligence for Science in Quantum, Atomistic, and Continuum Systems](https://mlsanity.com/paper/2307.08423/) (2023)
- [Reducing SO(3) Convolutions to SO(2) for Efficient Equivariant GNNs](https://mlsanity.com/paper/2302.03655/) (2023)
- [Exploiting Cyclic Symmetry in Convolutional Neural Networks](https://mlsanity.com/paper/1602.02660/) (2016)
- [DeePMD-kit: A deep learning package for many-body potential energy representation and molecular dynamics](https://mlsanity.com/paper/1712.03641/) (2017)
- [Unifying machine learning and quantum chemistry -- a deep neural network for molecular wavefunctions](https://mlsanity.com/paper/1906.10033/) (2019)
- [SE(3)-equivariant prediction of molecular wavefunctions and electronic densities](https://mlsanity.com/paper/2106.02347/) (2021)
- [Covariant Compositional Networks For Learning Graphs](https://mlsanity.com/paper/1801.02144/) (2018)
- [Spherical Channels for Modeling Atomic Interactions](https://mlsanity.com/paper/2206.14331/) (2022)
- [Efficient and Equivariant Graph Networks for Predicting Quantum Hamiltonian](https://mlsanity.com/paper/2306.04922/) (2023)

## Built on by (most cited first, 2 of 2)

- [TraceGrad: a Framework Learning Expressive SO(3)-equivariant Non-linear Representations for Electronic-Structure Hamiltonian Prediction](https://mlsanity.com/paper/2405.05722/) (2024)
- [Advancing Universal Deep Learning for Electronic-Structure Hamiltonian Prediction of Materials](https://mlsanity.com/paper/2509.19877/) (2025)

## Abstract (arXiv)

Deep learning for predicting the electronic-structure Hamiltonian of quantum systems necessitates satisfying the covariance laws, among which achieving SO(3)-equivariance without sacrificing the non-linear expressive capability of networks remains unsolved. To navigate the harmonization between equivariance and expressiveness, we propose a deep learning method synergizing two distinct categories of neural mechanisms as a two-stage encoding and regression framework. The first stage corresponds to group theory-based neural mechanisms with inherent SO(3)-equivariant properties prior to the parameter learning process, while the second stage is characterized by a non-linear 3D graph Transformer network we propose, featuring high capability on non-linear expressiveness. The novel combination lies in the point that, the first stage predicts baseline Hamiltonians with abundant SO(3)-equivariant features extracted, assisting the second stage in empirical learning of equivariance; and in turn, the second stage refines the first stage's output as a fine-grained prediction of Hamiltonians using powerful non-linear neural mappings, compensating for the intrinsic weakness on non-linear expressiveness capability of mechanisms in the first stage. Our method enables precise, generalizable predictions while capturing SO(3)-equivariance under rotational transformations, and achieves state-of-the-art performance in Hamiltonian prediction on six benchmark databases.