# Statistically Optimal Force Aggregation for Coarse-Graining Molecular Dynamics

arXiv: https://arxiv.org/abs/2302.07071 · HTML: https://mlsanity.com/paper/2302.07071/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 100 references that are in the map.
- First posted on arXiv 14 Feb 2023.

## Builds on (most cited first, 19 of 19)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Optimization Methods for Large-Scale Machine Learning](https://mlsanity.com/paper/1606.04838/) (2016)
- [SAGA: A Fast Incremental Gradient Method With Support for Non-Strongly Convex Composite Objectives](https://mlsanity.com/paper/1407.0202/) (2014)
- [SARAH: A Novel Method for Machine Learning Problems Using Stochastic Recursive Gradient](https://mlsanity.com/paper/1703.00102/) (2017)
- [TorchMD-NET: Equivariant Transformers for Neural Network based Molecular Potentials](https://mlsanity.com/paper/2202.02541/) (2022)
- [Minimizing Finite Sums with the Stochastic Average Gradient](https://mlsanity.com/paper/1309.2388/) (2013)
- [OSQP: An Operator Splitting Solver for Quadratic Programs](https://mlsanity.com/paper/1711.08013/) (2017)
- [Forces are not Enough: Benchmark and Critical Evaluation for Machine Learning Force Fields with Molecular Simulations](https://mlsanity.com/paper/2210.07237/) (2022)
- [Machine Learning of coarse-grained Molecular Dynamics Force Fields](https://mlsanity.com/paper/1812.01736/) (2018)
- [Coarse Graining Molecular Dynamics with Graph Neural Networks](https://mlsanity.com/paper/2007.11412/) (2020)
- [On the Ineffectiveness of Variance Reduced Optimization for Deep Learning](https://mlsanity.com/paper/1812.04529/) (2018)
- [Machine Learning Coarse-Grained Potentials of Protein Thermodynamics](https://mlsanity.com/paper/2212.07492/) (2022)
- [Flow-matching -- efficient coarse-graining of molecular dynamics without forces](https://mlsanity.com/paper/2203.11167/) (2022)
- [Coarse-Graining Auto-Encoders for Molecular Dynamics](https://mlsanity.com/paper/1812.02706/) (2018)
- [Learning neural network potentials from experimental data via Differentiable Trajectory Reweighting](https://mlsanity.com/paper/2106.01138/) (2021)
- [Machine Learning Implicit Solvation for Molecular Dynamics](https://mlsanity.com/paper/2106.07492/) (2021)
- [Contrastive Learning of Coarse-Grained Force Fields](https://mlsanity.com/paper/2205.10861/) (2022)
- [Predictive Coarse-Graining](https://mlsanity.com/paper/1605.08301/) (2016)
- [Developing Machine-Learned Potentials for Coarse-Grained Molecular Simulations: Challenges and Pitfalls](https://mlsanity.com/paper/2209.12948/) (2022)

## Built on by (most cited first, 9 of 9)

- [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)
- [Implicit Transfer Operator Learning: Multiple Time-Resolution Surrogates for Molecular Dynamics](https://mlsanity.com/paper/2305.18046/) (2023)
- [DiAMoNDBack: Diffusion-denoising Autoregressive Model for Non-Deterministic Backmapping of Cα Protein Traces](https://mlsanity.com/paper/2307.12451/) (2023)
- [AMARO: All Heavy-Atom Transferable Neural Network Potentials of Protein Thermodynamics](https://mlsanity.com/paper/2409.17852/) (2024)
- [Thermodynamic Transferability in Coarse-Grained Force Fields using Graph Neural Networks](https://mlsanity.com/paper/2406.12112/) (2024)
- [Universally applicable and tunable graph-based coarse-graining for Machine learning force fields](https://mlsanity.com/paper/2504.01973/) (2025)
- [Operator Forces For Coarse-Grained Molecular Dynamics](https://mlsanity.com/paper/2506.19628/) (2025)
- [Temperature transferable Machine Learned Coarse Grained model for proteins](https://mlsanity.com/paper/2606.14111/) (2026)

## Abstract (arXiv)

Machine-learned coarse-grained (CG) models have the potential for simulating large molecular complexes beyond what is possible with atomistic molecular dynamics. However, training accurate CG models remains a challenge. A widely used methodology for learning CG force-fields maps forces from all-atom molecular dynamics to the CG representation and matches them with a CG force-field on average. We show that there is flexibility in how to map all-atom forces to the CG representation, and that the most commonly used mapping methods are statistically inefficient and potentially even incorrect in the presence of constraints in the all-atom simulation. We define an optimization statement for force mappings and demonstrate that substantially improved CG force-fields can be learned from the same simulation data when using optimized force maps. The method is demonstrated on the miniproteins Chignolin and Tryptophan Cage and published as open-source code.