# FES-FM: Free Energy Surface Sampling via Reduced Flow Matching

arXiv: https://arxiv.org/abs/2605.00337 · HTML: https://mlsanity.com/paper/2605.00337/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 25 of its 45 references that are in the map.
- First posted on arXiv 1 May 2026.

## 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, 25 of 25)

- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [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 Flows: Exact Likelihood Generative Learning for Symmetric Densities](https://mlsanity.com/paper/2006.02425/) (2020)
- [Adjoint Matching: Fine-tuning Flow and Diffusion Generative Models with Memoryless Stochastic Optimal Control](https://mlsanity.com/paper/2409.08861/) (2024)
- [SE(3) diffusion model with application to protein backbone generation](https://mlsanity.com/paper/2302.02277/) (2023)
- [Path Integral Sampler: a stochastic control approach for sampling](https://mlsanity.com/paper/2111.15141/) (2021)
- [Learning Gradient Fields for Molecular Conformation Generation](https://mlsanity.com/paper/2105.03902/) (2021)
- [An optimal control perspective on diffusion-based generative modeling](https://mlsanity.com/paper/2211.01364/) (2022)
- [Iterated Denoising Energy Matching for Sampling from Boltzmann Densities](https://mlsanity.com/paper/2402.06121/) (2024)
- [Improved sampling via learned diffusions](https://mlsanity.com/paper/2307.01198/) (2023)
- [Adjoint Sampling: Highly Scalable Diffusion Samplers via Adjoint Matching](https://mlsanity.com/paper/2504.11713/) (2025)
- [NETS: A Non-Equilibrium Transport Sampler](https://mlsanity.com/paper/2410.02711/) (2024)
- [Target Score Matching](https://mlsanity.com/paper/2402.08667/) (2024)
- [Sequential Controlled Langevin Diffusions](https://mlsanity.com/paper/2412.07081/) (2024)
- [Scalable Equilibrium Sampling with Sequential Boltzmann Generators](https://mlsanity.com/paper/2502.18462/) (2025)
- [Monge-Amp`ere Flow for Generative Modeling](https://mlsanity.com/paper/1809.10188/) (2018)
- [Consistent Sampling and Simulation: Molecular Dynamics with Energy-Based Diffusion Models](https://mlsanity.com/paper/2506.17139/) (2025)
- [Dynamical Measure Transport and Neural PDE Solvers for Sampling](https://mlsanity.com/paper/2407.07873/) (2024)
- [Improving the Euclidean Diffusion Generation of Manifold Data by Mitigating Score Function Singularity](https://mlsanity.com/paper/2505.09922/) (2025)
- [FEAT: Free energy Estimators with Adaptive Transport](https://mlsanity.com/paper/2504.11516/) (2025)
- [Efficient Training of Energy-Based Models Using Jarzynski Equality](https://mlsanity.com/paper/2305.19414/) (2023)
- [Iterated Energy-based Flow Matching for Sampling from Boltzmann Densities](https://mlsanity.com/paper/2408.16249/) (2024)
- [Neural Sampling from Boltzmann Densities: Fisher-Rao Curves in the Wasserstein Geometry](https://mlsanity.com/paper/2410.03282/) (2024)
- [Energy based diffusion generator for efficient sampling of Boltzmann distributions](https://mlsanity.com/paper/2401.02080/) (2024)

## Abstract (arXiv)

Sampling the distribution of collective variables (CVs) and estimating the associated free energy surface are crucial problems in statistical physics, as they underpin a better understanding of chemical reactions and conformational transitions. Traditional methods usually rely on simulations in high-dimensional configuration space and project the resulting configurations onto the CV space. To improve sampling speed, we propose FES-FM, a reduced flow matching (FM) method for free energy surface (FES) sampling. We train a dynamical transport map in the CV space, thereby enabling direct sampling of CV distributions and reconstruction of the corresponding free energy surface. For many-particle systems, we construct a prior distribution based on the Hessian at a local minimum of the potential, which ensures both rotation-translation invariance and physically meaningful configurations. We evaluate the proposed method across a variety of potential functions and collective variables, including alanine dipeptide in implicit solvent as a molecular benchmark. Comparative experiments demonstrate that our approach significantly improves sampling speed while maintaining accuracy.