Hessian Matching for Machine-Learned Coarse-Grained Molecular Dynamics

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.