# Equivariant flow matching

arXiv: https://arxiv.org/abs/2306.15030 · HTML: https://mlsanity.com/paper/2306.15030/

## Key facts

- Cited by 135 papers in the map (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 44 of its 74 references that are in the map.
- First posted on arXiv 26 Jun 2023.

## Placed in

- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

## Builds on (most cited first, 30 of 44)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Analyzing and Improving the Image Quality of StyleGAN](https://mlsanity.com/paper/1912.04958/) (2019)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [Equivariant Diffusion for Molecule Generation in 3D](https://mlsanity.com/paper/2203.17003/) (2022)
- [GeoDiff: a Geometric Diffusion Model for Molecular Conformation Generation](https://mlsanity.com/paper/2203.02923/) (2022)
- [Semi-Supervised Learning with Deep Generative Models](https://mlsanity.com/paper/1406.5298/) (2014)
- [Equivariant message passing for the prediction of tensorial properties and molecular spectra](https://mlsanity.com/paper/2102.03150/) (2021)
- [GemNet: Universal Directional Graph Neural Networks for Molecules](https://mlsanity.com/paper/2106.08903/) (2021)
- [Learning from Protein Structure with Geometric Vector Perceptrons](https://mlsanity.com/paper/2009.01411/) (2020)
- [Multisample Flow Matching: Straightening Flows with Minibatch Couplings](https://mlsanity.com/paper/2304.14772/) (2023)
- [Equiformer: Equivariant Graph Attention Transformer for 3D Atomistic Graphs](https://mlsanity.com/paper/2206.11990/) (2022)
- [Torsional Diffusion for Molecular Conformer Generation](https://mlsanity.com/paper/2206.01729/) (2022)
- [Equivariant Flows: Exact Likelihood Generative Learning for Symmetric Densities](https://mlsanity.com/paper/2006.02425/) (2020)
- [How to train your neural ODE: the world of Jacobian and kinetic regularization](https://mlsanity.com/paper/2002.02798/) (2020)
- [E(n) Equivariant Normalizing Flows](https://mlsanity.com/paper/2105.09016/) (2021)
- [TrajectoryNet: A Dynamic Optimal Transport Network for Modeling Cellular Dynamics](https://mlsanity.com/paper/2002.04461/) (2020)
- [Stochastic Normalizing Flows](https://mlsanity.com/paper/2002.06707/) (2020)
- [OT-Flow: Fast and Accurate Continuous Normalizing Flows via Optimal Transport](https://mlsanity.com/paper/2006.00104/) (2020)
- [Flow Annealed Importance Sampling Bootstrap](https://mlsanity.com/paper/2208.01893/) (2022)
- [Equivariant flow-based sampling for lattice gauge theory](https://mlsanity.com/paper/2003.06413/) (2020)

## Built on by (most cited first, 30 of 135)

- [SE(3)-Stochastic Flow Matching for Protein Backbone Generation](https://mlsanity.com/paper/2310.02391/) (2023)
- [Iterated Denoising Energy Matching for Sampling from Boltzmann Densities](https://mlsanity.com/paper/2402.06121/) (2024)
- [Consistency Flow Matching: Defining Straight Flows with Velocity Consistency](https://mlsanity.com/paper/2407.02398/) (2024)
- [Stochastic interpolants with data-dependent couplings](https://mlsanity.com/paper/2310.03725/) (2023)
- [Adjoint Sampling: Highly Scalable Diffusion Samplers via Adjoint Matching](https://mlsanity.com/paper/2504.11713/) (2025)
- [Variational Flow Matching for Graph Generation](https://mlsanity.com/paper/2406.04843/) (2024)
- [Equivariant Flow Matching with Hybrid Probability Transport](https://mlsanity.com/paper/2312.07168/) (2023)
- [Mosaic-SDF for 3D Generative Models](https://mlsanity.com/paper/2312.09222/) (2023)
- [Categorical Flow Matching on Statistical Manifolds](https://mlsanity.com/paper/2405.16441/) (2024)
- [Transferable Boltzmann Generators](https://mlsanity.com/paper/2406.14426/) (2024)
- [OmniFlow: Any-to-Any Generation with Multi-Modal Rectified Flows](https://mlsanity.com/paper/2412.01169/) (2024)
- [Improved motif-scaffolding with SE(3) flow matching](https://mlsanity.com/paper/2401.04082/) (2024)
- [ET-Flow: Equivariant Flow-Matching for Molecular Conformer Generation](https://mlsanity.com/paper/2410.22388/) (2024)
- [Streaming Flow Policy: Simplifying diffusion/flow-matching policies by treating action trajectories as flow trajectories](https://mlsanity.com/paper/2505.21851/) (2025)
- [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)
- [Flow matching achieves almost minimax optimal convergence](https://mlsanity.com/paper/2405.20879/) (2024)
- [Free-form Flows: Make Any Architecture a Normalizing Flow](https://mlsanity.com/paper/2310.16624/) (2023)
- [Progressive Inference-Time Annealing of Diffusion Models for Sampling from Boltzmann Densities](https://mlsanity.com/paper/2506.16471/) (2025)
- [Consistent Sampling and Simulation: Molecular Dynamics with Energy-Based Diffusion Models](https://mlsanity.com/paper/2506.17139/) (2025)
- [FlowMol3: Flow Matching for 3D De Novo Small-Molecule Generation](https://mlsanity.com/paper/2508.12629/) (2025)
- [EdgeBench: Unveiling Scaling Laws of Learning from Real-World Environments](https://mlsanity.com/paper/2607.05155/) (2026)
- [FLOWR: Flow Matching for Structure-Aware De Novo, Interaction- and Fragment-Based Ligand Generation](https://mlsanity.com/paper/2504.10564/) (2025)
- [Not-So-Optimal Transport Flows for 3D Point Cloud Generation](https://mlsanity.com/paper/2502.12456/) (2025)
- [The Curse of Conditions: Analyzing and Improving Optimal Transport for Conditional Flow-Based Generation](https://mlsanity.com/paper/2503.10636/) (2025)
- [FEAT: Free energy Estimators with Adaptive Transport](https://mlsanity.com/paper/2504.11516/) (2025)
- [Reflected Flow Matching](https://mlsanity.com/paper/2405.16577/) (2024)
- [BNEM: A Boltzmann Sampler Based on Bootstrapped Noised Energy Matching](https://mlsanity.com/paper/2409.09787/) (2024)
- [Flow Matching Meets Biology and Life Science: A Survey](https://mlsanity.com/paper/2507.17731/) (2025)
- [Morphological Symmetries in Robotics](https://mlsanity.com/paper/2402.15552/) (2024)

## Abstract (arXiv)

Normalizing flows are a class of deep generative models that are especially interesting for modeling probability distributions in physics, where the exact likelihood of flows allows reweighting to known target energy functions and computing unbiased observables. For instance, Boltzmann generators tackle the long-standing sampling problem in statistical physics by training flows to produce equilibrium samples of many-body systems such as small molecules and proteins. To build effective models for such systems, it is crucial to incorporate the symmetries of the target energy into the model, which can be achieved by equivariant continuous normalizing flows (CNFs). However, CNFs can be computationally expensive to train and generate samples from, which has hampered their scalability and practical application. In this paper, we introduce equivariant flow matching, a new training objective for equivariant CNFs that is based on the recently proposed optimal transport flow matching. Equivariant flow matching exploits the physical symmetries of the target energy for efficient, simulation-free training of equivariant CNFs. We demonstrate the effectiveness of flow matching on rotation and permutation invariant many-particle systems and a small molecule, alanine dipeptide, where for the first time we obtain a Boltzmann generator with significant sampling efficiency without relying on tailored internal coordinate featurization. Our results show that the equivariant flow matching objective yields flows with shorter integration paths, improved sampling efficiency, and higher scalability compared to existing methods.