Hessian Matching for Machine-Learned Coarse-Grained Molecular Dynamics

2026-05-12

AI Summary

Current coarse-grained molecular dynamics neural potentials rely solely on force matching and lack constraints on the second-order curvature of the free energy landscape, limiting their accuracy and transferability. This work proposes an efficient curvature-aware supervision method that incorporates second-order information into the potential without explicitly constructing the full Hessian matrix, by matching stochastic Hessian-vector products. The approach innovatively decomposes the target coarse-grained Hessian into a projected all-atom Hessian and an online covariance correction term, and introduces an unbiased stochastic estimator to enable stable training. Evaluation on nine unseen fast-folding proteins shows that the method outperforms force-matching-only baselines in capturing slow dynamical modes for eight systems, with up to an 85% reduction in KL divergence along the slowest collective variable for the largest system.

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.

Problem

• coarse-grained molecular dynamics
• Hessian matching
• force matching
• free-energy surface curvature
• biomolecular simulation

Innovation

• Hessian matching
• coarse-grained molecular dynamics
• force matching
• stochastic Hessian-vector product
• free-energy curvature