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

arXiv: https://arxiv.org/abs/2603.04523 · HTML: https://mlsanity.com/paper/2603.04523/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 47 references that are in the map.
- First posted on arXiv 4 Mar 2026.

## Builds on (most cited first, 14 of 14)

- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentials](https://mlsanity.com/paper/2101.03164/) (2021)
- [Train faster, generalize better: Stability of stochastic gradient descent](https://mlsanity.com/paper/1509.01240/) (2015)
- [MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields](https://mlsanity.com/paper/2206.07697/) (2022)
- [Equivariant message passing for the prediction of tensorial properties and molecular spectra](https://mlsanity.com/paper/2102.03150/) (2021)
- [Sketching as a Tool for Numerical Linear Algebra](https://mlsanity.com/paper/1411.4357/) (2014)
- [Deep Potential Molecular Dynamics: a scalable model with the accuracy of quantum mechanics](https://mlsanity.com/paper/1707.09571/) (2017)
- [Machine Learning Force Fields](https://mlsanity.com/paper/2010.07067/) (2020)
- [Less is more: sampling chemical space with active learning](https://mlsanity.com/paper/1801.09319/) (2018)
- [Hierarchical modeling of molecular energies using a deep neural network](https://mlsanity.com/paper/1710.00017/) (2017)
- [Towards Fast, Specialized Machine Learning Force Fields: Distilling Foundation Models via Energy Hessians](https://mlsanity.com/paper/2501.09009/) (2025)
- [HIP: Hessian Interatomic Potentials without derivatives](https://mlsanity.com/paper/2509.21624/) (2025)
- [Hessian QM9: A quantum chemistry database of molecular Hessians in implicit solvents](https://mlsanity.com/paper/2408.08006/) (2024)
- [PFT: Phonon Fine-tuning for Machine Learned Interatomic Potentials](https://mlsanity.com/paper/2601.07742/) (2026)

## Built on by (most cited first, 4 of 4)

- [Hessian-informed machine learning interatomic potential towards bridging theory and experiments](https://mlsanity.com/paper/2603.25373/) (2026)
- [Hessian Matching for Machine-Learned Coarse-Grained Molecular Dynamics](https://mlsanity.com/paper/2605.12823/) (2026)
- [Hessian-Informed Conformation Augmentation for Potential Energy Surface Curvature in Machine Learning Interatomic Potentials](https://mlsanity.com/paper/2609.05233/) (2026)
- [Truncated automatic sparse differentiation for machine learning interatomic potentials](https://mlsanity.com/paper/2609.20510/) (2026)

## Abstract (arXiv)

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.