# Operator Forces For Coarse-Grained Molecular Dynamics

arXiv: https://arxiv.org/abs/2506.19628 · HTML: https://mlsanity.com/paper/2506.19628/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 41 of its 96 references that are in the map.
- First posted on arXiv 24 Jun 2025.

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

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
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- [GPT-4 Technical Report](https://mlsanity.com/paper/2303.08774/) (2023)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [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)
- [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(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [Sliced Score Matching: A Scalable Approach to Density and Score Estimation](https://mlsanity.com/paper/1905.07088/) (2019)
- [Equivariant Flows: Exact Likelihood Generative Learning for Symmetric Densities](https://mlsanity.com/paper/2006.02425/) (2020)
- [TorchMD-NET: Equivariant Transformers for Neural Network based Molecular Potentials](https://mlsanity.com/paper/2202.02541/) (2022)
- [E(n) Equivariant Normalizing Flows](https://mlsanity.com/paper/2105.09016/) (2021)
- [Equivariant flow matching](https://mlsanity.com/paper/2306.15030/) (2023)
- [Two for One: Diffusion Models and Force Fields for Coarse-Grained Molecular Dynamics](https://mlsanity.com/paper/2302.00600/) (2023)
- [Timewarp: Transferable Acceleration of Molecular Dynamics by Learning Time-Coarsened Dynamics](https://mlsanity.com/paper/2302.01170/) (2023)
- [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)
- [Transferable Boltzmann Generators](https://mlsanity.com/paper/2406.14426/) (2024)
- [Scalable Equilibrium Sampling with Sequential Boltzmann Generators](https://mlsanity.com/paper/2502.18462/) (2025)
- [Machine Learning Coarse-Grained Potentials of Protein Thermodynamics](https://mlsanity.com/paper/2212.07492/) (2022)
- [SE(3) Equivariant Augmented Coupling Flows](https://mlsanity.com/paper/2308.10364/) (2023)
- [Flow-matching -- efficient coarse-graining of molecular dynamics without forces](https://mlsanity.com/paper/2203.11167/) (2022)
- [Rigid Body Flows for Sampling Molecular Crystal Structures](https://mlsanity.com/paper/2301.11355/) (2023)

## Abstract (arXiv)

Coarse-grained (CG) molecular dynamics simulations extend the length and time scale of atomistic simulations by replacing groups of correlated atoms with CG beads. Machine-learned coarse-graining (MLCG) has recently emerged as a promising approach to construct highly accurate force fields for CG molecular dynamics. However, the calibration of MLCG force fields typically hinges on force matching, which demands extensive reference atomistic trajectories with corresponding force labels. In practice, atomistic forces are often not recorded, making traditional force matching infeasible on pre-existing datasets. Recently, noise-based kernels have been introduced to adapt force matching to the low-data regime, including situations in which reference atomistic forces are not present. While this approach produces force fields which recapitulate slow collective motion, it introduces significant local distortions due to the corrupting effects of the noise-based kernel. In this work, we introduce more general kernels based on normalizing flows that substantially reduce these local distortions while preserving global conformational accuracy. We demonstrate our method on small proteins, showing that flow-based kernels can generate high-quality CG forces solely from configurational samples.