Shoot from the HIP: Hessian Interatomic Potentials without derivatives

Machine learning interatomic potentials (MLIPs) learn the mapping from atomic positions to potential energy. The forces, the negative gradient of this energy, drive molecular dynamics and are readily obtained using automatic differentiation. Higher-order derivatives, most notably the Hessian, describe collective motion and allow the direct prediction of experimental observables, but are considered computationally inaccessible for large systems. We suggest a solution: in physical systems, interactions decay with distance, and most MLIPs build on this locality through message passing up to a finite receptive field. This implies both sparsity of higher-order derivatives and their decay with distance. This structure can be exploited using automatic sparse differentiation (ASD). We explain how to compute the sparsity pattern for MLIP derivatives and demonstrate that, for multiple foundation MLIPs, ASD computes full Hessians of large porous materials exactly, but with modest speedups at best. The larger gains come from truncated ASD: discarding small, but nonzero, Hessian entries between distant atoms yields order-of-magnitude speedups with negligible impact on predicted observables.

While machine-learning interatomic potentials (MLIPs) have successfully learned potential energy surfaces (PES) and atomic forces, many practical applications, such as vibrational analysis and transition state search, rely heavily on the PES Hessian. Yet standard MLIPs are trained on energy and forces alone, and existing methods that incorporate the Hessian into training objectives require architectural modifications and incur significant computational and memory overheads from higher-order backpropagation. To address these limitations, we propose two Hessian-derived data augmentation schemes: isotropic Gaussian displacement (\textbf{UniAug}) and normal mode-weighted displacement (\textbf{ModeAug}). Both methods utilize simple Taylor expansions, achieving effective augmentation without altering training objectives or extending the autograd graph. This allows seamless, plug-and-play integration with existing architectures and training pipelines. Comprehensive evaluations across non-equilibrium and equilibrium datasets demonstrate that our approach enhances model accuracy where reference forces are large while providing practical, task-specific guidelines.

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.