Hessian Matching for Machine-Learned Coarse-Grained Molecular Dynamics

TL;DR

This paper introduces a Hessian matching framework for coarse-grained molecular dynamics, improving the accuracy of neural potentials by incorporating second-order curvature information without full Hessian computation.

Contribution

The authors propose a novel stochastic Hessian-vector product matching method that enhances force matching for CG potentials, leading to better biomolecular simulation accuracy.

Findings

HVP matching outperforms force matching on 8 of 9 proteins in benchmarks.

Achieves up to 85% reduction in KL divergence along slow modes.

Enables more accurate and transferable CG potentials for biomolecular simulations.

Abstract

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 unconstrained. 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 construct an unbiased stochastic estimator of the Hessian-matching objective by using random probe vectors. We evaluate our method by comparing against force matching on a benchmark of nine fast-folding proteins unseen during training. HVP matching outperforms plain force matching on 8 of 9 proteins on slow-mode metrics, with reductions of up to 85% in the Kullback--Leibler divergence between the CG and reference distributions along the slowest collective mode of the largest protein. Our results demonstrate that higher-order physical supervision is a practical path to more accurate and transferable CG potentials for biomolecular simulation.