# Flow-matching -- efficient coarse-graining of molecular dynamics without forces

arXiv: https://arxiv.org/abs/2203.11167 · HTML: https://mlsanity.com/paper/2203.11167/

## Key facts

- Cited by 36 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 30 of its 98 references that are in the map.
- First posted on arXiv 21 Mar 2022.

## 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 30)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [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)
- [Sigmoid-Weighted Linear Units for Neural Network Function Approximation in Reinforcement Learning](https://mlsanity.com/paper/1702.03118/) (2017)
- [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)
- [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019)
- [Directional Message Passing for Molecular Graphs](https://mlsanity.com/paper/2003.03123/) (2020)
- [Sliced Score Matching: A Scalable Approach to Density and Score Estimation](https://mlsanity.com/paper/1905.07088/) (2019)
- [How to Train Your Energy-Based Models](https://mlsanity.com/paper/2101.03288/) (2021)
- [Stochastic Normalizing Flows](https://mlsanity.com/paper/2002.06707/) (2020)
- [Normalizing Flows on Tori and Spheres](https://mlsanity.com/paper/2002.02428/) (2020)
- [SurVAE Flows: Surjections to Bridge the Gap between VAEs and Flows](https://mlsanity.com/paper/2007.02731/) (2020)
- [Machine Learning of coarse-grained Molecular Dynamics Force Fields](https://mlsanity.com/paper/1812.01736/) (2018)
- [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)
- [Coarse Graining Molecular Dynamics with Graph Neural Networks](https://mlsanity.com/paper/2007.11412/) (2020)
- [Smooth Normalizing Flows](https://mlsanity.com/paper/2110.00351/) (2021)
- [VFlow: More Expressive Generative Flows with Variational Data Augmentation](https://mlsanity.com/paper/2002.09741/) (2020)
- [Deeptime: a Python library for machine learning dynamical models from time series data](https://mlsanity.com/paper/2110.15013/) (2021)
- [Coarse-Graining Auto-Encoders for Molecular Dynamics](https://mlsanity.com/paper/1812.02706/) (2018)
- [Generative Coarse-Graining of Molecular Conformations](https://mlsanity.com/paper/2201.12176/) (2022)
- [Machine Learning Implicit Solvation for Molecular Dynamics](https://mlsanity.com/paper/2106.07492/) (2021)
- [Neural Canonical Transformation with Symplectic Flows](https://mlsanity.com/paper/1910.00024/) (2019)
- [Ensemble Learning of Coarse-Grained Molecular Dynamics Force Fields with a Kernel Approach](https://mlsanity.com/paper/2005.01851/) (2020)
- [Manifold Density Estimation via Generalized Dequantization](https://mlsanity.com/paper/2102.07143/) (2021)

## Built on by (most cited first, 30 of 36)

- [Equivariant flow matching](https://mlsanity.com/paper/2306.15030/) (2023)
- [Variational Flow Matching for Graph Generation](https://mlsanity.com/paper/2406.04843/) (2024)
- [Two for One: Diffusion Models and Force Fields for Coarse-Grained Molecular Dynamics](https://mlsanity.com/paper/2302.00600/) (2023)
- [Transferable Boltzmann Generators](https://mlsanity.com/paper/2406.14426/) (2024)
- [SE(3) Equivariant Augmented Coupling Flows](https://mlsanity.com/paper/2308.10364/) (2023)
- [Str2Str: A Score-based Framework for Zero-shot Protein Conformation Sampling](https://mlsanity.com/paper/2306.03117/) (2023)
- [Rigid Body Flows for Sampling Molecular Crystal Structures](https://mlsanity.com/paper/2301.11355/) (2023)
- [Navigating protein landscapes with a machine-learned transferable coarse-grained model](https://mlsanity.com/paper/2310.18278/) (2023)
- [Consistent Sampling and Simulation: Molecular Dynamics with Energy-Based Diffusion Models](https://mlsanity.com/paper/2506.17139/) (2025)
- [EquiJump: Protein Dynamics Simulation via SO(3)-Equivariant Stochastic Interpolants](https://mlsanity.com/paper/2410.09667/) (2024)
- [Exponential Family Variational Flow Matching for Tabular Data Generation](https://mlsanity.com/paper/2506.05940/) (2025)
- [Implicit Transfer Operator Learning: Multiple Time-Resolution Surrogates for Molecular Dynamics](https://mlsanity.com/paper/2305.18046/) (2023)
- [chemtrain: Learning Deep Potential Models via Automatic Differentiation and Statistical Physics](https://mlsanity.com/paper/2408.15852/) (2024)
- [DiAMoNDBack: Diffusion-denoising Autoregressive Model for Non-Deterministic Backmapping of Cα Protein Traces](https://mlsanity.com/paper/2307.12451/) (2023)
- [Scalable Normalizing Flows Enable Boltzmann Generators for Macromolecules](https://mlsanity.com/paper/2401.04246/) (2024)
- [Statistically Optimal Force Aggregation for Coarse-Graining Molecular Dynamics](https://mlsanity.com/paper/2302.07071/) (2023)
- [Dynamic PDB: A New Dataset and a SE(3) Model Extension by Integrating Dynamic Behaviors and Physical Properties in Protein Structures](https://mlsanity.com/paper/2408.12413/) (2024)
- [Controlled Generation with Equivariant Variational Flow Matching](https://mlsanity.com/paper/2506.18340/) (2025)
- [BoltzNCE: Learning Likelihoods for Boltzmann Generation with Stochastic Interpolants and Noise Contrastive Estimation](https://mlsanity.com/paper/2507.00846/) (2025)
- [Latent Representation and Simulation of Markov Processes via Time-Lagged Information Bottleneck](https://mlsanity.com/paper/2309.07200/) (2023)
- [Conditional Normalizing Flows for Active Learning of Coarse-Grained Molecular Representations](https://mlsanity.com/paper/2402.01195/) (2024)
- [Top-down machine learning of coarse-grained protein force-fields](https://mlsanity.com/paper/2306.11375/) (2023)
- [Purrception: Variational Flow Matching for Vector-Quantized Image Generation](https://mlsanity.com/paper/2510.01478/) (2025)
- [AlphaFolding: 4D Diffusion for Dynamic Protein Structure Prediction with Reference and Motion Guidance](https://mlsanity.com/paper/2408.12419/) (2024)
- [Beyond Ensembles: Simulating All-Atom Protein Dynamics in a Learned Latent Space](https://mlsanity.com/paper/2509.02196/) (2025)
- [Enhanced Sampling for Efficient Learning of Coarse-Grained Machine Learning Potentials](https://mlsanity.com/paper/2510.11148/) (2025)
- [Energy-Based Coarse-Graining in Molecular Dynamics: A Flow-Based Framework without Data](https://mlsanity.com/paper/2504.20940/) (2025)
- [Internal-Coordinate Density Modelling of Protein Structure: Covariance Matters](https://mlsanity.com/paper/2302.13711/) (2023)
- [Sifting through the Noise: A Survey of Diffusion Probabilistic Models and Their Applications to Biomolecules](https://mlsanity.com/paper/2406.01622/) (2024)
- [AI-based Methods for Simulating, Sampling, and Predicting Protein Ensembles](https://mlsanity.com/paper/2509.17224/) (2025)

## Abstract (arXiv)

Coarse-grained (CG) molecular simulations have become a standard tool to study molecular processes on time- and length-scales inaccessible to all-atom simulations. Parameterizing CG force fields to match all-atom simulations has mainly relied on force-matching or relative entropy minimization, which require many samples from costly simulations with all-atom or CG resolutions, respectively. Here we present flow-matching, a new training method for CG force fields that combines the advantages of both methods by leveraging normalizing flows, a generative deep learning method. Flow-matching first trains a normalizing flow to represent the CG probability density, which is equivalent to minimizing the relative entropy without requiring iterative CG simulations. Subsequently, the flow generates samples and forces according to the learned distribution in order to train the desired CG free energy model via force matching. Even without requiring forces from the all-atom simulations, flow-matching outperforms classical force-matching by an order of magnitude in terms of data efficiency, and produces CG models that can capture the folding and unfolding transitions of small proteins.