Projected Hessian Learning: Fast Curvature Supervision for Accurate Machine-Learning Interatomic Potentials

TL;DR

Projected Hessian Learning (PHL) introduces a scalable method for incorporating curvature information into machine-learning interatomic potentials using Hessian-vector products, achieving second-order accuracy without quadratic memory costs.

Contribution

PHL provides a novel stochastic trace-based framework for curvature-informed training that scales efficiently, enabling accurate second-order potential training without explicit Hessian computation.

Findings

HVP-based schemes match full Hessian training accuracy

PHL achieves >24x speedup over full Hessian methods

Randomized probes outperform fixed probes for complex geometries

Abstract

The Hessian matrix (second derivatives) encodes far richer local curvature of the potential energy surface than energies and forces alone. However, training machine-learning interatomic potentials (MLIPs) with full Hessians is often impractical because explicitly forming and storing Hessian matrices scales quadratically in cost and memory. We introduce Projected Hessian Learning (PHL), a scalable second-order training framework that injects curvature information using only Hessian-vector products (HVPs). Rather than constructing the Hessian, PHL projects curvature along stochastic probe directions and uses an unbiased stochastic trace-based loss with favorable system-size scaling, enabling curvature-informed training without quadratic memory growth. We benchmark PHL on a chemically diverse dataset of reactants, products, transition states, intrinsic reaction coordinates, and normal-mode sampled geometries computed at omegaB97XD/6-31G(d). We compare energy-force training (E-F), two HVP-based schemes (E-F-HVP with one-hot or randomized probes), and full energy-force-Hessian training (E-F-H). With randomized probes per minibatch, both HVP schemes match full-Hessian training in energy, force, and Hessian accuracy while delivering >24x epoch speedups for the small molecular systems studied. In a fixed-probe regime with one HVP per molecule, randomized projections consistently outperform one-column probing, especially for far-from-equilibrium geometries. Overall, PHL replaces explicit Hessian supervision with force-complexity curvature training, retaining most second-order accuracy gains while scaling to larger, more complex molecular systems.