# Rigid Body Flows for Sampling Molecular Crystal Structures

arXiv: https://arxiv.org/abs/2301.11355 · HTML: https://mlsanity.com/paper/2301.11355/

## Key facts

- Cited by 27 papers in the map (data through 14 Oct 2026).
- Builds on 41 of its 83 references that are in the map.
- First posted on arXiv 26 Jan 2023.

## Builds on (most cited first, 30 of 41)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (2016)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentials](https://mlsanity.com/paper/2101.03164/) (2021)
- [MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields](https://mlsanity.com/paper/2206.07697/) (2022)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [Equivariant Flows: Exact Likelihood Generative Learning for Symmetric Densities](https://mlsanity.com/paper/2006.02425/) (2020)
- [Efficient and Modular Implicit Differentiation](https://mlsanity.com/paper/2105.15183/) (2021)
- [E(n) Equivariant Normalizing Flows](https://mlsanity.com/paper/2105.09016/) (2021)
- [Stochastic Normalizing Flows](https://mlsanity.com/paper/2002.06707/) (2020)
- [Neural Importance Sampling](https://mlsanity.com/paper/1808.03856/) (2018)
- [Flow-based generative models for Markov chain Monte Carlo in lattice field theory](https://mlsanity.com/paper/1904.12072/) (2019)
- [Riemannian Continuous Normalizing Flows](https://mlsanity.com/paper/2006.10605/) (2020)
- [Flow Annealed Importance Sampling Bootstrap](https://mlsanity.com/paper/2208.01893/) (2022)
- [Normalizing Flows on Tori and Spheres](https://mlsanity.com/paper/2002.02428/) (2020)
- [Convex Potential Flows: Universal Probability Distributions with Optimal Transport and Convex Optimization](https://mlsanity.com/paper/2012.05942/) (2020)
- [Learning Neural Generative Dynamics for Molecular Conformation Generation](https://mlsanity.com/paper/2102.10240/) (2021)
- [Sampling using SU(N) gauge equivariant flows](https://mlsanity.com/paper/2008.05456/) (2020)
- [Neural Manifold Ordinary Differential Equations](https://mlsanity.com/paper/2006.10254/) (2020)
- [SurVAE Flows: Surjections to Bridge the Gap between VAEs and Flows](https://mlsanity.com/paper/2007.02731/) (2020)
- [Normalizing Flows on Riemannian Manifolds](https://mlsanity.com/paper/1611.02304/) (2016)
- [Targeted free energy estimation via learned mappings](https://mlsanity.com/paper/2002.04913/) (2020)
- [Augmented Normalizing Flows: Bridging the Gap Between Generative Flows and Latent Variable Models](https://mlsanity.com/paper/2002.07101/) (2020)
- [Implicit-PDF: Non-Parametric Representation of Probability Distributions on the Rotation Manifold](https://mlsanity.com/paper/2106.05965/) (2021)
- [Matching Normalizing Flows and Probability Paths on Manifolds](https://mlsanity.com/paper/2207.04711/) (2022)

## Built on by (most cited first, 27 of 27)

- [Equivariant flow matching](https://mlsanity.com/paper/2306.15030/) (2023)
- [Iterated Denoising Energy Matching for Sampling from Boltzmann Densities](https://mlsanity.com/paper/2402.06121/) (2024)
- [Adjoint Sampling: Highly Scalable Diffusion Samplers via Adjoint Matching](https://mlsanity.com/paper/2504.11713/) (2025)
- [Timewarp: Transferable Acceleration of Molecular Dynamics by Learning Time-Coarsened Dynamics](https://mlsanity.com/paper/2302.01170/) (2023)
- [Clifford Group Equivariant Neural Networks](https://mlsanity.com/paper/2305.11141/) (2023)
- [Transferable Boltzmann Generators](https://mlsanity.com/paper/2406.14426/) (2024)
- [Space Group Informed Transformer for Crystalline Materials Generation](https://mlsanity.com/paper/2403.15734/) (2024)
- [Scalable Equilibrium Sampling with Sequential Boltzmann Generators](https://mlsanity.com/paper/2502.18462/) (2025)
- [SE(3) Equivariant Augmented Coupling Flows](https://mlsanity.com/paper/2308.10364/) (2023)
- [Efficient mapping of phase diagrams with conditional Boltzmann Generators](https://mlsanity.com/paper/2406.12378/) (2024)
- [FALCON: Few-step Accurate Likelihoods for Continuous Flows](https://mlsanity.com/paper/2512.09914/) (2025)
- [Amortized Sampling with Transferable Normalizing Flows](https://mlsanity.com/paper/2508.18175/) (2025)
- [ReQFlow: Rectified Quaternion Flow for Efficient and High-Quality Protein Backbone Generation](https://mlsanity.com/paper/2502.14637/) (2025)
- [Scalable Boltzmann Generators for equilibrium sampling of large-scale materials](https://mlsanity.com/paper/2509.25486/) (2025)
- [UniSim: A Unified Simulator for Time-Coarsened Dynamics of Biomolecules](https://mlsanity.com/paper/2506.03157/) (2025)
- [Space Group Conditional Flow Matching](https://mlsanity.com/paper/2509.23822/) (2025)
- [Force-Guided Bridge Matching for Full-Atom Time-Coarsened Dynamics of Peptides](https://mlsanity.com/paper/2408.15126/) (2024)
- [Generative Marginalization Models](https://mlsanity.com/paper/2310.12920/) (2023)
- [Topological Obstructions and How to Avoid Them](https://mlsanity.com/paper/2312.07529/) (2023)
- [Generative Models for Crystalline Materials](https://mlsanity.com/paper/2511.22652/) (2025)
- [Multi-Type Point Cloud Autoencoder: A Complete Equivariant Embedding for Molecule Conformation and Pose](https://mlsanity.com/paper/2405.13791/) (2024)
- [Importance Weighted Score Matching for Diffusion Samplers with Enhanced Mode Coverage](https://mlsanity.com/paper/2505.19431/) (2025)
- [Charting Empirical Laws for LLM Fine-Tuning in Scientific Multi-Discipline Learning](https://mlsanity.com/paper/2602.11215/) (2026)
- [Accurate Free Energy Estimations of Molecular Systems Via Flow-based Targeted Free Energy Perturbation](https://mlsanity.com/paper/2302.11855/) (2023)
- [Flow Perturbation to Accelerate Unbiased Sampling of Boltzmann distribution](https://mlsanity.com/paper/2407.10666/) (2024)
- [Operator Forces For Coarse-Grained Molecular Dynamics](https://mlsanity.com/paper/2506.19628/) (2025)
- [FRAGMENTA: End-to-end Fragmentation-based Generative Model with Agentic Tuning for Drug Lead Optimization](https://mlsanity.com/paper/2511.20510/) (2025)

## Abstract (arXiv)

Normalizing flows (NF) are a class of powerful generative models that have gained popularity in recent years due to their ability to model complex distributions with high flexibility and expressiveness. In this work, we introduce a new type of normalizing flow that is tailored for modeling positions and orientations of multiple objects in three-dimensional space, such as molecules in a crystal. Our approach is based on two key ideas: first, we define smooth and expressive flows on the group of unit quaternions, which allows us to capture the continuous rotational motion of rigid bodies; second, we use the double cover property of unit quaternions to define a proper density on the rotation group. This ensures that our model can be trained using standard likelihood-based methods or variational inference with respect to a thermodynamic target density. We evaluate the method by training Boltzmann generators for two molecular examples, namely the multi-modal density of a tetrahedral system in an external field and the ice XI phase in the TIP4P water model. Our flows can be combined with flows operating on the internal degrees of freedom of molecules and constitute an important step towards the modeling of distributions of many interacting molecules.