Projected Hessian learning: fast curvature supervision for accurate machine-learning interatomic potentials
- 1Los Alamos National Laboratory
- 2Michigan State University
- 3NVIDIA Corp
Abstract
The Hessian matrix of second derivatives contains substantially richer information about the local geometry of the potential energy surface than energies and forces alone. Although incorporating full Hessians into machine-learning interatomic potential (MLIP) training can significantly improve accuracy and robustness, the quadratic computational and memory cost of explicitly constructing and storing Hessian matrices has limited its practical use. Here, we introduce Projected Hessian Learning (PHL), a scalable second-order training framework that incorporates curvature information using only Hessian–vector products (HVPs). By avoiding explicit Hessian construction and instead projecting curvature along stochastic probe directions, PHL reduces the cost of second-derivative supervision to near force-level complexity. The resulting dense-probe trace estimator is unbiased and exhibits favorable scaling with system size, enabling curvature-informed training without quadratic memory growth. We evaluated various training approaches on a chemically diverse dataset of reactants, products, transition states, intrinsic reaction coordinates, and normal-mode sampled geometries generated at the theoretical level ω B97XD/6-31G(d). Four training schemes are considered: energy-force training (E-F), two HVP-based approaches (E-F-HVP with one-hot or dense Gaussian vectors) and energy-force-Hessian training (E-F-H). When probe vectors are randomized for each minibatch, both HVP-based approaches achieve energy, force, and Hessian accuracy comparable to full Hessian training while providing more than 24 × speedups per epoch for the small molecular systems studied here. In a more realistic fixed-vector regime with only one HVP per molecule, dense Gaussian projections consistently outperform one-column probing, particularly for far-from-equilibrium geometries. Paired follow-up training with Rademacher probes gives no statistically significant change in model accuracy relative to the original Gaussian choice, although Rademacher probes have lower single-probe estimator variance and bounded component magnitudes. Overall, PHL effectively replaces explicit Hessian supervision with force-complexity curvature training, retaining most of the accuracy benefits of full second-order methods while enabling scalable MLIP development for larger and more complex molecular systems.
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BibTeX
@article{rodriguez2026projected,
title = {{Projected Hessian learning: fast curvature supervision for accurate machine-learning interatomic potentials}},
author = {Austin Rodriguez and Justin Smith and Sakib Matin and Nicholas Lubbers and Kipton M. Barros and José L. Mendoza-Cortés},
journal = {Machine Learning Science and Technology},
year = {2026},
month = sep,
doi = {10.1088/2632-2153/aeac9b},
url = {https://doi.org/10.1088/2632-2153/aeac9b},
note = {CC BY},
}
RIS
TY - JOUR
TI - Projected Hessian learning: fast curvature supervision for accurate machine-learning interatomic potentials
AU - Austin Rodriguez
AU - Justin Smith
AU - Sakib Matin
AU - Nicholas Lubbers
AU - Kipton M. Barros
AU - José L. Mendoza-Cortés
PY - 2026
DA - 2026/09/25
JO - Machine Learning Science and Technology
DO - 10.1088/2632-2153/aeac9b
UR - https://doi.org/10.1088/2632-2153/aeac9b
AB - The Hessian matrix of second derivatives contains substantially richer information about the local geometry of the potential energy surface than energies and forces alone. Although incorporating full Hessians into machine-learning interatomic potential (MLIP) training can significantly improve accuracy and robustness, the quadratic computational and memory cost of explicitly constructing and storing Hessian matrices has limited its practical use. Here, we introduce Projected Hessian Learning (PHL), a scalable second-order training framework that incorporates curvature information using only Hessian–vector products (HVPs). By avoiding explicit Hessian construction and instead projecting curvature along stochastic probe directions, PHL reduces the cost of second-derivative supervision to near force-level complexity. The resulting dense-probe trace estimator is unbiased and exhibits favorable scaling with system size, enabling curvature-informed training without quadratic memory growth. We evaluated various training approaches on a chemically diverse dataset of reactants, products, transition states, intrinsic reaction coordinates, and normal-mode sampled geometries generated at the theoretical level ω B97XD/6-31G(d). Four training schemes are considered: energy-force training (E-F), two HVP-based approaches (E-F-HVP with one-hot or dense Gaussian vectors) and energy-force-Hessian training (E-F-H). When probe vectors are randomized for each minibatch, both HVP-based approaches achieve energy, force, and Hessian accuracy comparable to full Hessian training while providing more than 24 × speedups per epoch for the small molecular systems studied here. In a more realistic fixed-vector regime with only one HVP per molecule, dense Gaussian projections consistently outperform one-column probing, particularly for far-from-equilibrium geometries. Paired follow-up training with Rademacher probes gives no statistically significant change in model accuracy relative to the original Gaussian choice, although Rademacher probes have lower single-probe estimator variance and bounded component magnitudes. Overall, PHL effectively replaces explicit Hessian supervision with force-complexity curvature training, retaining most of the accuracy benefits of full second-order methods while enabling scalable MLIP development for larger and more complex molecular systems.
SN - 2632-2153
ER -