# Hessian Matching for Machine-Learned Coarse-Grained Molecular Dynamics

arXiv: https://arxiv.org/abs/2605.12823 · HTML: https://mlsanity.com/paper/2605.12823/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 38 references that are in the map.
- First posted on arXiv 12 May 2026.

## Builds on (most cited first, 11 of 11)

- [Optimizing Neural Networks with Kronecker-factored Approximate Curvature](https://mlsanity.com/paper/1503.05671/) (2015)
- [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)
- [TorchMD: A deep learning framework for molecular simulations](https://mlsanity.com/paper/2012.12106/) (2020)
- [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)
- [Navigating protein landscapes with a machine-learned transferable coarse-grained model](https://mlsanity.com/paper/2310.18278/) (2023)
- [TorchMD-Net 2.0: Fast Neural Network Potentials for Molecular Simulations](https://mlsanity.com/paper/2402.17660/) (2024)
- [HIP: Hessian Interatomic Potentials without derivatives](https://mlsanity.com/paper/2509.21624/) (2025)
- [Projected Hessian Learning: Fast Curvature Supervision for Accurate Machine-Learning Interatomic Potentials](https://mlsanity.com/paper/2603.04523/) (2026)
- [A Standardized Benchmark for Machine-Learned Molecular Dynamics using Weighted Ensemble Sampling](https://mlsanity.com/paper/2510.17187/) (2025)

## Abstract (arXiv)

Coarse-grained (CG) molecular dynamics enables simulations of atomic systems such as biomolecules at timescales inaccessible to all-atom (AA) methods, but existing CG neural potentials trained via force matching capture only the gradient of the free-energy surface, leaving its curvature unsupervised. We introduce a framework that augments force matching with stochastic Hessian-vector product (HVP) matching, instilling second-order curvature information into CG potentials without constructing the full Hessian. We derive a decomposition of the target CG Hessian into a model-independent projected AA Hessian, precomputed once before training, and a model-dependent covariance correction computed online at negligible cost. We then construct an unbiased stochastic estimator of the Hessian-matching objective by using random probe vectors. We evaluate our method by training on a benchmark set of nine fast-folding proteins and gauging agreement of the learned free-energy landscapes with reference data along the directions of slowest collective motion using time-lagged independent component analysis. HVP matching improves slow-mode accuracy over force matching for all but one of our benchmark proteins. It also sharpens local structure, reducing bond-length error on eight of nine proteins, and in several cases by an order of magnitude. Our results demonstrate that higher-order physical supervision is a practical path to more accurate CG potentials for biomolecular simulation.