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

• URL: http://arxiv.org/abs/2603.04523v1
• Date: Wed, 04 Mar 2026 19:09:16 GMT
• Title: Projected Hessian Learning: Fast Curvature Supervision for Accurate Machine-Learning Interatomic Potentials
• Abstract summary: Projected Hessian Learning (PHL) is a scalable second-order training framework that injects curvature information using only Hessian-vector products (HVPs)<n>We benchmark PHL on a chemically diverse dataset of reactants, products transition states, and normal-mode sampled geometries computed at omegaB97XD/6-31G(d)<n>With randomized probes per minibatch, both HVP schemes match full-Hessian training in energy, force, and Hessian accuracy while delivering >24x speedups for the small molecular systems studied.
• Score: 3.0325110391148264
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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