# Multi-Scale Protein Structure Modelling with Geometric Graph U-Nets

arXiv: https://arxiv.org/abs/2512.06752 · HTML: https://mlsanity.com/paper/2512.06752/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 29 of its 45 references that are in the map.
- First posted on arXiv 7 Dec 2025.

## Builds on (most cited first, 29 of 29)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [PointNet++: Deep Hierarchical Feature Learning on Point Sets in a Metric Space](https://mlsanity.com/paper/1706.02413/) (2017)
- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [Weisfeiler and Leman Go Neural: Higher-order Graph Neural Networks](https://mlsanity.com/paper/1810.02244/) (2018)
- [Hierarchical Graph Representation Learning with Differentiable Pooling](https://mlsanity.com/paper/1806.08804/) (2018)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [On the Bottleneck of Graph Neural Networks and its Practical Implications](https://mlsanity.com/paper/2006.05205/) (2020)
- [SchNet: A continuous-filter convolutional neural network for modeling quantum interactions](https://mlsanity.com/paper/1706.08566/) (2017)
- [Learning from Protein Structure with Geometric Vector Perceptrons](https://mlsanity.com/paper/2009.01411/) (2020)
- [On the Expressive Power of Deep Neural Networks](https://mlsanity.com/paper/1606.05336/) (2016)
- [Fast and Uncertainty-Aware Directional Message Passing for Non-Equilibrium Molecules](https://mlsanity.com/paper/2011.14115/) (2020)
- [Spherical Message Passing for 3D Graph Networks](https://mlsanity.com/paper/2102.05013/) (2021)
- [On Over-Squashing in Message Passing Neural Networks: The Impact of Width, Depth, and Topology](https://mlsanity.com/paper/2302.02941/) (2023)
- [Protein Representation Learning by Geometric Structure Pretraining](https://mlsanity.com/paper/2203.06125/) (2022)
- [ASAP: Adaptive Structure Aware Pooling for Learning Hierarchical Graph Representations](https://mlsanity.com/paper/1911.07979/) (2019)
- [Artificial Intelligence for Science in Quantum, Atomistic, and Continuum Systems](https://mlsanity.com/paper/2307.08423/) (2023)
- [On the Expressive Power of Geometric Graph Neural Networks](https://mlsanity.com/paper/2301.09308/) (2023)
- [Understanding Pooling in Graph Neural Networks](https://mlsanity.com/paper/2110.05292/) (2021)
- [DeepSF: deep convolutional neural network for mapping protein sequences to folds](https://mlsanity.com/paper/1706.01010/) (2017)
- [The expressive power of pooling in Graph Neural Networks](https://mlsanity.com/paper/2304.01575/) (2023)
- [Transformers are Graph Neural Networks](https://mlsanity.com/paper/2506.22084/) (2025)
- [Geometry-Complete Perceptron Networks for 3D Molecular Graphs](https://mlsanity.com/paper/2211.02504/) (2022)
- [Evaluating representation learning on the protein structure universe](https://mlsanity.com/paper/2406.13864/) (2024)
- [Hierarchical Adaptive Pooling by Capturing High-order Dependency for Graph Representation Learning](https://mlsanity.com/paper/2104.05960/) (2021)
- [VN-EGNN: E(3)-Equivariant Graph Neural Networks with Virtual Nodes Enhance Protein Binding Site Identification](https://mlsanity.com/paper/2404.07194/) (2024)

## Abstract (arXiv)

Geometric Graph Neural Networks (GNNs) and Transformers have become state-of-the-art for learning from 3D protein structures. However, their reliance on message passing prevents them from capturing the hierarchical interactions that govern protein function, such as global domains and long-range allosteric regulation. In this work, we argue that the network architecture itself should mirror this biological hierarchy. We introduce Geometric Graph U-Nets, a new class of models that learn multi-scale representations by recursively coarsening and refining the protein graph. We prove that this hierarchical design can theoretically more expressive than standard Geometric GNNs. Empirically, on the task of protein fold classification, Geometric U-Nets substantially outperform invariant and equivariant baselines, demonstrating their ability to learn the global structural patterns that define protein folds. Our work provides a principled foundation for designing geometric deep learning architectures that can learn the multi-scale structure of biomolecules.