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REVIEW 4 major objections 4 minor 44 references

HORM: A Large Scale Molecular Hessian Database for Optimizing Reactive Machine Learning Interatomic Potentials

T0 review · 4 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The paper claims that 1.84 million Hessians, plus a row-sampled Hessian loss, are what reactive machine-learning potentials need to find transition states reliably.

desk verdict A useful reactive Hessian dataset and consistent evidence that Hessian supervision helps, but the 200x TS-search headline is fragile. read the letter →

arxiv 2505.12447 v1 pith:ZIO4UZTO submitted 2025-05-18 physics.chem-ph physics.comp-ph

Taoyong Cui , Yunhong Han , Haojun Jia , Chenru Duan , Qiyuan Zhao This is my paper · ORCID
classification physics.chem-phphysics.comp-ph
keywords Hessiandatabasemachinelearninginteratomicpotentialstransitionstatesearchsecond-orderderivativesreactivemoleculardynamicsstochasticrowsamplingomegaB97x/6-31G(d)direct-forcemodels
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper introduces HORM, a dataset of 1.84 million Hessian matrices for reactive molecular geometries at the $\omega$B97X/6-31G(d) level, and argues that training machine-learning interatomic potentials with second-derivative supervision makes them substantially better at finding transition states. The authors show that adding a Hessian loss term, computed cheaply by randomly sampling a few rows of the Hessian, improves Hessian prediction, restores near-symmetry to force-derived Hessians, and lifts the number of correctly recovered transition states, most dramatically for direct-force models, with EquiformerV2 going from 3 to 684 intended transition states. A sympathetic reader would care because transition-state searches are the bottleneck in automated reaction exploration, and this is a practical recipe for making reactive MLIPs usable for that task.

What carries the argument

HORM itself is the load-bearing object: 1,836,206 Hessian matrices recomputed at $\omega$B97X/6-31G(d) from Transition1x and RGD1 geometries, spanning non-equilibrium regions of the potential energy surface. The accompanying training method augments the standard energy-force loss with a Hessian-matching term in which a small random subset of Hessian rows (one for autograd models, two for direct-force models) is compared against the derivative of the model's predicted forces, computed by batched vector-Jacobian products so the cost is $O(s)$ rather than $O(N^2)$. This targeted supervision is what lets the models learn curvature and recover symmetric Hessians without paying the full computational price.

What would settle it

Run the same trained E-F and E-F-H models through an independent, transparent transition-state search pipeline, such as NEB or GSM followed by eigenvector-following refinement with IRC verification, and compare intended-TS counts; if E-F-H does not clearly outperform E-F, or if E-F already recovers hundreds of intended TSs, the reported 3-to-684 result will not reproduce.

Watch

Extended reading notes

Core claim

The paper's central claim is that explicit Hessian supervision during MLIP training, enabled by a dataset large enough to provide off-equilibrium second derivatives, is what makes reactive machine-learning potentials reliable for transition-state optimization. On the HORM-Transition1x validation set, adding the Hessian loss cuts Hessian mean absolute error by up to 97% (EquiformerV2: 2.231 to 0.075 eV/Å$^2$) and eigenvalue error by up to 99%, while in the end-to-end TS search benchmark the same model's number of intended transition states rises from 3 under energy-force training to 684 under energy-force-Hessian training. The authors interpret this as evidence that second-order information enforces force consistency and correct local curvature, overcoming a known limitation of direct-force architectures.

Load-bearing premise

The headline TS-search gain rests on the separate end-to-end TS search workflow cited as [29] and its IRC-based definition of an intended TS; if that workflow's success criteria are biased, or if the energy-force baseline is essentially a failed model, the 200x comparison is not a fair measure.

Editorial extensions

If this is right

  • Direct-force MLIPs, which are fast but normally produce non-conservative forces and asymmetric Hessians, become viable for transition-state search once trained with Hessian supervision.
  • Training on HORM improves out-of-distribution generalization: on the RGD1 subset, EquiformerV2 energy MAE drops by 45% and Hessian MAE by 93% relative to energy-force-only training.
  • Autograd-based models also gain, with in-distribution Hessian and eigenvalue MAEs reduced by up to 59% and 78%, even though their force errors barely change.
  • Barrier-height prediction improves by up to 10%, while TS geometry RMSD changes little, indicating that the Hessian supervision primarily corrects local curvature rather than the overall geometry.
  • The dataset and row-sampling loss provide a template for training future reactive MLIPs that need second-order information without paying the full $O(N^2)$ Hessian cost.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The 3-to-684 jump suggests the energy-force-trained EquiformerV2 was essentially failing at the TS search task, so the '200 times' ratio may overstate the practical gain over a reasonable baseline; a fairer comparison would match training budgets and report success rates per reaction.
  • Because HORM covers mostly C/H/O/N systems with under eight heavy atoms, the observed gains are a proof-of-concept for small-molecule reactivity; extending the same recipe to P/S/halogens or larger fragments is the natural next test and the authors say they are working toward it.
  • Stochastic row sampling with just one or two Hessian rows per structure implies that full-Hessian supervision is unnecessary; an adaptive scheme that samples rows near the largest predicted curvature could make the method even cheaper while retaining the symmetry-enforcing benefit.
  • The Hessian dataset could also serve as a benchmark for evaluating whether a model learns physically consistent curvature, independent of its downstream TS search performance.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. This manuscript introduces HORM, a dataset of 1,836,206 Hessian matrices computed at the ωB97X/6-31G(d) level for reactive configurations sampled from Transition1x (1,776,206 geometries) and RGD1 (60,000 geometries). The authors propose an energy-force-Hessian (E-F-H) training loss with stochastic row sampling and compare it with standard energy-force (E-F) training for AlphaNet, LEFTNet (autograd and direct-force variants), and EquiformerV2. They report consistent reductions in Hessian and eigenvalue MAE on the in-distribution Transition1x validation set and the out-of-distribution RGD1 set, reduced Hessian asymmetry for direct-force models, and large gains in an end-to-end transition state search benchmark, most prominently an increase from 3 to 684 intended TSs for EquiformerV2. The paper claims up to 200 times higher TS search success rates and up to 63% reduction in Hessian MAE.

Significance. The dataset fills a real gap: off-equilibrium Hessians of reactive systems are scarce, and HORM is an order-of-magnitude larger resource than Hessian-QM9, with diverse non-equilibrium geometries. The central methodological result—that adding Hessian supervision with only one or two sampled rows per Hessian improves second-order accuracy, Hessian symmetry, and TS-related performance across four model variants and on out-of-distribution data—is plausible and, if confirmed, practically valuable. The paper deserves credit for benchmarking multiple architectures, including the OOD split, and for reporting Hessian asymmetry errors. However, the headline TS-search claim is not yet robustly supported because it depends on a near-zero baseline and an external, unpublished workflow.

major comments (4)
  1. [Section 5.2, Figs. 2a-b] The headline 'up to 200 times' improvement rests on an E-F baseline of only 3 intended TSs for EquiformerV2. With such a small denominator, the ratio is highly sensitive to convergence thresholds, a few borderline cases, or stochasticity. The paper also does not report the total number of test reactions, so the reader cannot judge whether 684 represents a high absolute success rate. In addition, the evaluation relies entirely on the authors' end-to-end TS search workflow in reference [29], an unpublished ChemRxiv preprint by overlapping authors; the workflow's initial-guess generation, GSM settings, convergence criteria, and IRC verification thresholds are not described in this manuscript. Please provide the test-set size, describe the workflow (or validate it independently), and report success counts with a stable, non-degenerate baseline before using the 200x ratio as a headline claim.
  2. [Abstract vs. Tables 1 and 2] The abstract states 'up to 63% reduction in the Hessian mean absolute error,' but Tables 1 and 2 report reductions of 97% (EquiformerV2, HORM-Transition1x validation) and 93% (EquiformerV2, HORM-RGD1) in Hessian MAE relative to the E-F baseline, and Section 5.1 explicitly states those numbers. Please reconcile the abstract with the tables; if 63% refers to a different metric, model subset, or aggregation, say so explicitly.
  3. [Equations (2)-(3) and Table A.2] The loss weights α, β, and γ are never specified. Table A.2 reports NHR, learning rate, and batch size but omits the weights that control the trade-off among energy, force, and Hessian terms. Because the main conclusions depend on these weights, omitting them hinders reproduction and sensitivity analysis. Relatedly, the claim that sampling one row (autograd models) or two rows (direct-force models) per Hessian per epoch is sufficient is an assumption; no ablation over NHR or γ is provided. Please report the weights and add a small sensitivity study, even for a single representative model.
  4. [Tables 1 and A.1] The claim in Section 5.1 that Hessian supervision improves performance across 'nearly all evaluation metrics' is not fully supported by the tables: LEFTNet-df force MAE on the Transition1x validation set worsens from 0.029 to 0.044 eV/Å under E-F-H training (Table 1), and the corresponding OOD force MAE shows only a marginal change. This is a reporting inconsistency worth addressing, since the text presents force behavior as essentially neutral or improved.
minor comments (4)
  1. [Table captions, Tables 1-2] The word 'parathensis' appears in the caption text; it should be 'parentheses'.
  2. [Section 4, Eq. (3)] The notation H^(j)_i and F^(j)_φ is introduced in the text but could be defined immediately after the equation for clarity, particularly the meaning of the superscript j as the selected Hessian row or force entry.
  3. [Data availability] No data or code availability statement is provided. Since HORM is the central contribution, a repository URL or a clear statement about how to obtain the dataset and training/evaluation code should be added.
  4. [Section 5.2] The TS search evaluation reports no uncertainty or multiple-seed statistics for the intended-TS counts; given the 3->684 ratio is a headline result, at least a repeated-run or bootstrap-style measure would help.

Circularity Check

1 steps flagged · score 4.0 of 10

TS-search headline rests on an unverified self-cited workflow, while the Hessian-error results are independent.

  1. self citation load bearing [Section 5.2 (Transition State Search Performance), with the intended-TS definition in Section 2.3]
    "To evaluate the practical capabilities of reactive MLIPs in realistic TS search scenarios, we assess their performance using our recently developed end-to-end TS search workflow.[29] ... Among all evaluated metrics, the number of intended TSs showed the most substantial improvement, with EquiformerV2 increasing from just 3 intended TSs under E–F to 684 under E–F–H (Fig. 2a–b)."

    The headline 'up to 200 times' TS-search gain is measured entirely by the authors' own unpublished workflow, reference [29], a ChemRxiv preprint with overlapping authorship. That workflow is not described in this paper: its IRC verification thresholds, initial-guess generation, and convergence criteria are external to the manuscript. The 'intended TS' count and the 3-to-684 ratio are outputs of that unverified self-cited pipeline, so the central TS-search improvement claim is not independently testable from the paper's own equations or data. The Hessian MAE improvements in Tables 1–2 are independent DFT-referenced results and are not circular, which limits the overall score.

full rationale

The core dataset and Hessian-informed training comparison are not circular: the E-F-H loss (Eq. 3) is evaluated against held-out Transition1x validation and OOD RGD1 data, with reference Hessians recomputed at the ωB97x/6-31G(d) level, and the loss weights and stochastic row-sampling counts are hyperparameters rather than quantities fitted to the target results. No fitted parameter is renamed as a prediction, no known result is merely relabeled, and no uniqueness theorem is imported. The only substantial circularity burden is the TS-search evaluation, which relies on the authors' own unpublished workflow [29] to define and count 'intended TSs'; because that workflow is not described or independently validated in this paper, the 200x improvement claim reduces to an unverified self-citation. The abstract's 'up to 63%' Hessian MAE reduction is inconsistent with the 97% and 93% reductions in Tables 1 and 2, but this is a numerical inconsistency rather than a circular argument. Overall, the paper has independent and valuable Hessian-prediction content, so the score is 4 rather than higher.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

The central comparison E-F vs E-F-H does not require fitting any constants to the headline result, but the reported gains are contingent on hand-chosen loss weights, a small fixed number of sampled Hessian rows, and an external self-cited TS workflow. These are assumptions rather than fitted parameters, and they are not stress-tested in the paper.

free parameters (5)
  • Energy loss weight alpha = not stated
    Chosen by hand in Eq. 2; no values or sensitivity analysis reported.
  • Force loss weight beta = not stated
    Chosen by hand in Eq. 2; no values reported.
  • Hessian loss weight gamma = not stated
    Chosen by hand in Eq. 3; no values or tuning curve reported.
  • Number of Hessian rows sampled (NHR) = 1 for autograd-based models, 2 for direct-force models
    Set in Table A.2; the paper does not test sensitivity to this choice, yet it controls the strength of Hessian supervision.
  • Learning rate and batch size = See Table A.2 (1e-4 to 3e-4; 32 to 128)
    Standard hyperparameters, not fitted to the benchmark, but they affect all models equally.
assumptions (4)
  • domain assumption omega B97x/6-31G(d) DFT provides reference energies, forces, and Hessians accurate enough for TS-oriented MLIP training.
    Used throughout Sec 3.1 for recomputation with GPU4PYSCF; no comparison to higher-level theory.
  • domain assumption Geometries sampled from Transition1x and RGD1 are representative of the reactive chemical space HORM claims to cover.
    Sec 3.1 describes 20%/5% sampling from Transition1x and up to 15 IRC points per reaction from 80,000 RGD1 reactions; representativeness is assessed only via t-SNE and property distributions.
  • ad hoc to paper Sampling one or two rows per Hessian per epoch is sufficient to learn accurate Hessian structure.
    Introduced in Sec 4 and fixed in Table A.2; no ablation varying NHR is provided, making the training strategy's success depend on an untested choice.
  • domain assumption The end-to-end TS search workflow of reference [29] reliably identifies intended TSs.
    Sec 5.2 uses this workflow for all TS metrics; the workflow is a ChemRxiv preprint by overlapping authors and is not described or validated in this paper.

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Cite this review

Pith. "Pith review of HORM: A Large Scale Molecular Hessian Database for Optimizing Reactive Machine Learning Interatomic Potentials." pith.science (2026). https://pith.science/paper/ZIO4UZTO

@misc{pith2026250512447,
  author       = {Pith},
  title        = {Pith review of: HORM: A Large Scale Molecular Hessian Database for Optimizing Reactive Machine Learning Interatomic Potentials},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZIO4UZTO}},
  note         = {Machine review of arXiv:2505.12447}
}
abstract

Transition state (TS) characterization is central to computational reaction modeling, yet conventional approaches depend on expensive density functional theory (DFT) calculations, limiting their scalability. Machine learning interatomic potentials (MLIPs) have emerged as a promising approach to accelerate TS searches by approximating quantum-level accuracy at a fraction of the cost. However, most MLIPs are primarily designed for energy and force prediction, thus their capacity to accurately estimate Hessians, which are crucial for TS optimization, remains constrained by limited training data and inadequate learning strategies. This work introduces the Hessian dataset for Optimizing Reactive MLIP (HORM), the largest quantum chemistry Hessian database dedicated to reactive systems, comprising 1.84 million Hessian matrices computed at the $\omega$B97x/6-31G(d) level of theory. To effectively leverage this dataset, we adopt a Hessian-informed training strategy that incorporates stochastic row sampling, which addresses the dramatically increased cost and complexity of incorporating second-order information into MLIPs. Various MLIP architectures and force prediction schemes trained on HORM demonstrate up to 63% reduction in the Hessian mean absolute error and up to 200 times increase in TS search compared to models trained without Hessian information. These results highlight how HORM addresses critical data and methodological gaps, enabling the development of more accurate and robust reactive MLIPs for large-scale reaction network exploration.

Figures

Figures reproduced from arXiv: 2505.12447 by the authors.

Figure 1
Figure 1. Molecular structure and property distributions of Hessian datasets. (a–c) t-SNE visual [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Comparison of MLIP performance in transition state search. (a-b) Number of reactions [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Radar plots comparing the performance of E-F and E-F-H models across different MLIPs. [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.