# Equivariant Neural Belief Propagation

arXiv: https://arxiv.org/abs/2606.06344 · HTML: https://mlsanity.com/paper/2606.06344/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 26 references that are in the map.
- First posted on arXiv 4 Jun 2026.

## Builds on (most cited first, 15 of 15)

- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentials](https://mlsanity.com/paper/2101.03164/) (2021)
- [SchNet: A continuous-filter convolutional neural network for modeling quantum interactions](https://mlsanity.com/paper/1706.08566/) (2017)
- [Directional Message Passing for Molecular Graphs](https://mlsanity.com/paper/2003.03123/) (2020)
- [GeoDiff: a Geometric Diffusion Model for Molecular Conformation Generation](https://mlsanity.com/paper/2203.02923/) (2022)
- [Equivariant message passing for the prediction of tensorial properties and molecular spectra](https://mlsanity.com/paper/2102.03150/) (2021)
- [Torsional Diffusion for Molecular Conformer Generation](https://mlsanity.com/paper/2206.01729/) (2022)
- [GEOM: Energy-annotated molecular conformations for property prediction and molecular generation](https://mlsanity.com/paper/2006.05531/) (2020)
- [Geometric and Physical Quantities Improve E(3) Equivariant Message Passing](https://mlsanity.com/paper/2110.02905/) (2021)
- [Expectation Propagation for approximate Bayesian inference](https://mlsanity.com/paper/1301.2294/) (2013)
- [GeoMol: Torsional Geometric Generation of Molecular 3D Conformer Ensembles](https://mlsanity.com/paper/2106.07802/) (2021)
- [Provably Strict Generalisation Benefit for Equivariant Models](https://mlsanity.com/paper/2102.10333/) (2021)
- [Loopy Belief Propagation for Approximate Inference: An Empirical Study](https://mlsanity.com/paper/1301.6725/) (2013)
- [Robust Differentiable SVD](https://mlsanity.com/paper/2104.03821/) (2021)

## Abstract (arXiv)

Probabilistic inference over spatially embedded variables requires beliefs that respect SE(3) symmetry, yet existing equivariant networks produce only scalars and vectors -- not the rank-2 precision tensors needed for anisotropic uncertainty, and single-component messages collapse multi-modal energy landscapes to physically meaningless averages. We introduce Equivariant Neural Belief Propagation (ENBP), a factor-graph framework whose messages are equivariant Gaussian mixture models with sufficient statistics that transform exactly under SE(3). Rank-2 precision matrices are synthesised via equivariant outer products, ingested through differentiable spectral decomposition, and kept tractable by a greedy KL-based mixture reduction that provably commutes with SE(3). On GEOM-QM9 and GEOM-Drugs, ENBP achieves 98.9% conformational coverage at 0.090 mathringA error with sub-second latency -- over 100× faster than diffusion baselines at higher accuracy. On multi-body robotic inference, vanilla loopy BP diverges at 15+ agents while ENBP converges with near-zero collision rates and machine-precision equivariance error (~10-7 vs. 10-1 for augmented baselines).