Hessian-informed machine learning interatomic potential towards bridging theory and experiments

Bangchen Yin; Jian Ouyang; Zhen Fan; Kailai Lin; Hanshi Hu; Dingshun Lv; Weiluo Ren; Hai Xiao

cs.LGMar 26, 2026

Hessian-informed machine learning interatomic potential towards bridging theory and experiments

Authors: Bangchen Yin, Jian Ouyang, Zhen Fan, Kailai Lin, Hanshi Hu, Dingshun Lv, Weiluo Ren, Hai Xiao,

Organizations: ByteDance Seed · Tsinghua University · Peking University

Abstract

Local curvature of potential energy surfaces is critical for predicting certain experimental observables of molecules and materials from first principles, yet it remains far beyond reach for complex systems. In this work, we introduce a Hessian-informed Machine Learning Interatomic Potential (Hi-MLIP) that captures such curvature reliably, thereby enabling accurate analysis of associated thermodynamic and kinetic phenomena. To make Hessian supervision practically viable, we develop a highly efficient training protocol, termed Hessian INformed Training (HINT), achieving two to four orders of magnitude reduction for the requirement of expensive Hessian labels. HINT integrates critical techniques, including Hessian pre-training, configuration sampling, curriculum learning and stochastic projection Hessian loss. Enabled by HINT, Hi-MLIP significantly improves transition-state search and brings Gibbs free-energy predictions close to chemical accuracy especially in data-scarce regimes. Our framework also enables accurate treatment of strongly anharmonic hydrides, reproducing phonon renormalization and superconducting critical temperatures in close agreement with experiment while bypassing the computational bottleneck of anharmonic calculations. These results establish a practical route to enhancing curvature awareness of machine learning interatomic potentials, bridging simulation and experimental observables across a wide range of systems.

Figures & tables

Figure 1: Framework of the Hessian-informed training (HINT) protocol enables accurate prediction of thermodynamic and kinetic properties. (a) Efficient data preparation. Hessian pre-training leverages large-scale low-fidelity (xTB) Hessian data to faithfully learn the global PES curvature. High-fidelity (DFT) configurations are then sampled via the weighted-local density sampling in SOAP descriptor space or energy-based ranking, where retaining only the top 5% of configurations by Hessian label achieves comparable performance to the full dataset. (b) Efficient training process. A curriculum learning strategy dynamically reweights the Energy/Force/Hessian labels over training epochs. The stochastic projected Hessian loss, based on the Hutchinson estimation with random $\pm 1$ vectors, achieves $\mathcal{O}(N)$ training efficiency. (c) Hi-MLIP architecture. The HINT protocol augments a base MLIP with Hessian supervision and yields a Hi-MLIP. Force and Hessian predictions from the Hi-MLIP are obtained by successive analytical differentiation of the potential energy, ensuring physical consistency and improved accuracy over the base MLIP model. (d) Applications enabled by Hi-MLIP, including PES traversal for transition state searches, free energy and other thermodynamic property prediction, and phonon anharmonicity calculation in hydrides under pressure.

Figure 2: Data efficiency and predictive accuracy of the Hessian-informed training protocol on molecular reactions. (a, b) Label efficiency of the protocol, showing MAEs for (a) energy/forces and (b) Hessian/eigenvalues with increasing Hessian training samples. Horizontal dashed lines denote the full-dataset baseline ( $\sim$ 1.7M configurations). (c) Ablation study comparing loss functions and pre-training strategies (PT) using ${10^{5}}$ Hessian samples. (d) Success rates of transition state (TS) searches trained on varying energy-based subsets (high-energy Top 5–20% vs. low-energy Bot 20–40%), comparing models trained with (EFH) and without (EF) Hessian labels. (e) TS search success rate versus Hessian data ratio within the Top 5% high-energy subset. (f) Generalization performance on free energy prediction. The average MAE for 124 randomly selected reactant/product (R/P) and transition state (TS) structures are compared across different models. The best model achieves the highest accuracy. The 5% EF + 0.1% Hess (0.0005% Hess of full dataset) model demonstrates remarkable data efficiency, outperforming the HORM baselines and approaching chemical accuracy (black dashed line, 0.043 eV).

Figure 3: Anharmonic effects on the dynamic stability of hydrosulfide and palladium hydrides. (a) Workflow for predicting anharmonic dynamic matrix ( $\Phi$ ) using the SSCHA framework enhanced by Hessian-informed training protocol (HINT). Red boxes represent DFT calculations, blue boxes represent HINT part, and green box represents SSCHA part. (b) Energy variation of the H 3 S structure as a function of the displacement order parameter (where H atoms move slightly toward S atoms, indicating a second-order phase transition) under different pressures. The calculations are performed using the AlphaNet model with Hessian fine-tuning (without Hessian fine-tuning for inset). Solid dark dots indicate the local minima. (c) Harmonic and anharmonic phonon dispersions of the $Im\bar{3}m$ phase of H 3 S at 150 GPa. The gray dashed lines represent the harmonic results calculated by DFT, while the red solid lines represent the anharmonic results optimized by SSCHA using Hi-MLIP. (d) , (e) , (f) Harmonic and anharmonic phonon dispersions with the projected phonon density of states (DOS) for PdH, PdD, and PdT. For DOS, dashed and solid lines represent harmonic and anharmonic calculations, respectively. The blue and red filled regions denote the contributions from the Pd atoms and the H/D/T isotopes.

Figure 4: Efficient evaluation of phonon anharmonicity and superconducting critical temperatures via Hi-MLIP. (a) Temperature dependence of the superconducting gap ( $\Delta$ ) for PdH, PdD, and PdT, calculated by solving the Migdal-Eliashberg (ME) equations. Open diamonds with dashed lines indicate harmonic results, while solid circles with solid lines denote anharmonic results. The inset provides a magnified view of the low-temperature region for PdH, PdD and PdT. (b) Superconducting critical temperature ( $T_{c}$ ) for LaH 10 at 200 GPa (red), H 3 S at 150 GPa (blue), CaH 6 at 180 GPa (brown) and PdH at 1 atm (green). Open and solid symbols correspond to anharmonic and harmonic results, respectively. The dashed gray line indicates perfect agreement between theory and experiments [ 44 , 25 , 66 , 23 ] . (c) Comparison of computational efficiency between the Hi-MLIP approach and the pure DFT approach (estimated) for anharmonic calculations across four hydrides. (d) The pie charts illustrate the percentage of time consumed by each computational step for $\text{H}{\vphantom{\text{X}}}_{\smash[t]{\text{3}}}\text{S}$ .

Machine Learning Interatomic Potentials

Machine learning interatomic potentials predict atomic energies and forces for fast simulations of molecules and materials.

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

Austin Rodriguez, Justin S. Smith, Sakib Matin, Nicholas Lubbers+2

Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, United States. · Department of Chemical Engineering & Materials Science, Michigan State University, East Lansing, Michigan 48824, United States. · NVIDIA Corp., 2788 San Tomas Expy, Santa Clara, California 95051, United States.+3

Machine Learning Interatomic Potentials Second-Order Optimization

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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.

Abstract

Machine Learning Interatomic Potentials

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