Machine learning the computational cost of quantum chemistry

Guido Falk von Rudorff; Max Schwilk; O. Anatole von Lilienfeld; Stefan Heinen

REVIEW 3 major objections 6 minor 1 cited by

Machine learning the computational cost of quantum chemistry

T0 review · 3 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Quantum machine learning models can learn how long a quantum chemistry calculation will run from molecular structure alone, and packing jobs with those predictions cuts CPU-time overhead by 10 to 90 percent.

desk verdict A genuinely new and mostly solid demonstration that quantum-chemistry wall times are learnable from molecular structure; the scheduling-efficiency headline rests on a simulator with idealized assumptions and is softer than the abstract suggests. read the letter →

arxiv 1908.06714 v2 pith:EP672L5Z submitted 2019-08-19 physics.chem-ph physics.comp-ph

Stefan Heinen , Max Schwilk , Guido Falk von Rudorff , O. Anatole von Lilienfeld This is my paper · ORCID

classification physics.chem-phphysics.comp-ph

keywords quantummachinelearningjobschedulingwalltimepredictionkernelridgeregressioncomputationalcostmodelingchemistryfirst-fitdecreasinghigh-performancecomputing

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

The paper tries to establish that the computational cost of routine quantum chemistry jobs is a learnable, smoothly varying function of molecular structure. It trains kernel ridge regression models on wall-time labels for single-point, geometry-optimization, and transition-state calculations across thousands of organic molecules, and shows that out-of-sample prediction error falls systematically as the training set grows. If true, this matters because high-performance computing clusters currently treat all jobs of one type as identical, wasting CPU time and energy; accurate per-molecule run-time estimates turn scheduling into a bin-packing problem. The authors report that replacing indiscriminate scheduling with estimates from these models cuts CPU-time overhead by 10 to 90 percent across seven task types after training on only thousands of molecules.

What carries the argument

The engine is kernel ridge regression applied to molecular representations: wall time is normalized by electron count, described by either a two-body bond-count representation or a one-, two-, and three-body atomic descriptor, and learned with a Laplacian or Gaussian kernel. The learned absolute run-time estimates are then sorted in decreasing order and packed into compute jobs with a first-fit-decreasing heuristic, so idle time left by long jobs is filled by shorter ones. A secondary mechanism is step splitting for geometry optimizations: because the first optimization step costs about 20 percent more than later steps, modeling it separately improves both the timing model and the scheduler.

What would settle it

Run the published seven task sets on a production cluster that preempts jobs and has load-dependent queue delays, then compare CPU overhead when jobs are ordered by the learned run-time estimates against the conventional job-array approach; if overhead does not drop, the scheduling claim fails. A second check is whether the FLOP-count model, presented as a noise-free cost measure, transfers to a different CPU architecture within the same error bounds.

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Extended reading notes

Core claim

The central claim is that wall time is a molecular quasi-property: from the initial geometry and composition of a molecule, a kernel machine can predict how long a specified electronic-structure calculation will run, accurately enough to improve scheduling. Evidence is given for seven tasks spanning single-point calculations, geometry optimizations, and transition-state searches, with five levels of electronic-structure theory and thousands of closed- and open-shell organic molecules. At the largest training sizes, normalized mean absolute errors range from about 1.3 percent for single-point coupled-cluster timings to roughly 27 to 48 percent for geometry optimizations, and the learned timings, fed into a first-fit-decreasing scheduler, reduce CPU overhead by 10 to 90 percent relative to job arrays or job steps.

Load-bearing premise

The reported savings assume that once a calculation starts it runs to completion without interruption, that starting a new job costs 30 seconds, and that every job waits exactly one hour in the queue; clusters with preemption, variable waiting times, or busy input-output systems may not see those savings.

Editorial extensions

If this is right

Single-point, geometry-optimization, and transition-state wall times are all learnable from structure alone; the optimization tasks are harder but still accurate enough for scheduling.
Training sets of a few thousand molecules suffice to beat the baseline assumption that every calculation of a given type takes the mean run time.
Accurate run-time estimates let clusters pack jobs in decreasing order, roughly halving overhead at large core counts and allowing much larger parallel campaigns at the same overhead.
For geometry optimizations, predicting individual step durations lets a scheduler release resources early instead of starting a step that cannot finish, cutting overhead by up to 90 percent at small wall-time limits.

Reading between the lines

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

The same learned-cost view could extend to other expensive workflows, such as periodic solid-state calculations or ab initio molecular dynamics, where runtime also depends on structure and on the path taken.
Predicted run times could be used online to detect abnormal jobs: a large gap between predicted and observed duration would flag I/O contention, convergence failure, or a misconfigured input.
If runtime is truly a molecular quasi-property, cost models could guide generative molecular design by preferring cheaper candidate molecules, not just by scheduling a fixed campaign.
The reported gains depend on a simulator with fixed startup and queue delays; on real clusters with load-dependent queuing the savings could be larger if backfilling becomes easier, or smaller if preemption breaks the packing.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The thing to know about this paper: the core learning result is real. The authors treat wall time as a quasi-property, use kernel ridge regression with BoB and FCHL representations on seven distinct quantum-chemistry tasks (single points, geometry optimizations, transition states, at theory levels from B3LYP to MRCI+Q), and show out-of-sample errors that decay systematically with training set size, down to 1.3–40% at the largest N. That is genuinely new in the QML literature, and the data generation alone is substantial: thousands of timings across Molpro and ORCA, plus a FLOP-count control experiment that gives a cleaner target than wall time and confirms the learnability.

What the paper does well: it frames computational cost as a molecular quasi-property and shows that established representations capture it. The toy-system study is a useful sanity check; the kernel PCA analysis gives a plausible reason why FCHL beats BoB; and the code and most data are shipped, which helps others build on it.

Where it gets soft: the scheduling half. The headline 10–90% CPU-time overhead reductions are not measured on a cluster. They are outputs of a simulator that assumes idempotent, uninterruptible tasks, a fixed 30-second startup, and a one-hour queue delay for every job (Section 3.2.2). Real clusters have preemption, variable queue delays, and I/O contention that cannot be filtered away at scheduling time. The authors also drop runs with wall/CPU ratios above task-specific thresholds up to 50% (Section 4.2.4), which makes sense for building a clean learning target but means the scheduling experiments run on a sanitized workload. So the conclusion's 'demonstrably used compute clusters more efficiently' is too strong; what they demonstrated is that a simulator with those assumptions produces better packing.

Smaller issues: no error bars on the learning curves or overhead reductions; the choice of FCHL as the scheduling representation is made after seeing test-set errors (mild selection); and the carbene data set is only 'available upon request,' which is a reproducibility wrinkle. None of these sink the core result.

Bottom line: the learnability claim holds up. The scheduling gains are plausible but not yet proven in production. This paper deserves a serious referee who will push for a sensitivity analysis or a real scheduler trace. I would accept it for peer review, and I would cite it for the wall-time learning result, not for the efficiency numbers.

Referee Report

3 major / 6 minor

Summary. The paper introduces kernel ridge regression (KRR) models of computational cost for quantum chemistry, predicting wall times, CPU times, and FLOP counts for seven tasks spanning single-point, geometry-optimization, and transition-state calculations, plus a 2D toy optimization problem. Molecular inputs are represented with Bag of Bonds and FCHL fingerprints, and the authors report learning curves with normalized mean absolute errors, reaching roughly 1.3–48% MAE at the largest training sizes. The predicted absolute wall times are then fed into a first-fit-decreasing job scheduling simulator, and the paper claims reductions in CPU-time overhead of 10–90% relative to conventional job-array and job-step packing. The manuscript makes code, data, and a GitHub repository available.

Significance. If the scheduling claims survive closer scrutiny, this is a useful and novel application of QML to a practical HPC problem, and one of the first demonstrations that computational cost can be treated as a learnable molecular quasi-property across chemical space. The paper's strengths include systematically decaying learning curves, a broad set of tasks and levels of theory, the use of FLOP counts as a cleaner target, and openly available timings and scripts. The main weakness is that the headline efficiency gains are outputs of an idealized simulator, coupled with data-cleaning choices and test-set model selection that may bias the reported gains. The QML accuracy results themselves are honest and reproducible, but the translation from normalized MAEs to production scheduling savings is not yet fully supported.

major comments (3)

[§3.2.2 and §4.3.1, Figs. 9–10] The headline 10–90% CPU-time overhead reductions are simulator outputs, not cluster measurements. The simulator assumes idempotent uninterruptible tasks, a fixed 30-second startup cost, and a fixed one-hour queue for every job, and it does not model preemption, stochastic queue delays, or I/O contention. The conclusion that the authors 'demonstrably used compute clusters more efficiently' therefore overstates what the evidence shows. Please validate the scheduling claims on a real scheduler trace, or at minimum add a sensitivity analysis over queue-delay distributions, startup costs, and preemption behavior, and revise the wording to 'simulated' rather than 'demonstrably'.
[§4.2.4 and Fig. 3] Runs with wall-time-to-CPU-time ratios above task-specific thresholds of 3%, 5%, 10%, 30%, and 50% are excluded as 'heavy I/O' before training and evaluation. At scheduling time this ratio is not known before a job executes, so a production scheduler cannot apply the same filter. As a result, the simulator evaluates the QML scheduler on a cleaned distribution that is not available in practice. Please report how many jobs are excluded per task, and show that the learning curves and overhead reductions survive when the excluded jobs are kept or otherwise modeled, since the reported 10–90% range may be optimistically biased.
[§4.3.1 and Table 2] The representation used for scheduling is selected as the one with the lowest MAE at the maximum training size, where those MAEs are computed on the held-out test set. This constitutes test-set model selection, which inflates the reported generalization accuracy and, consequently, the downstream scheduling gains. Please replace this procedure with nested cross-validation or a pre-registered representation choice, or at least report the scheduling results for both BoB and FCHL models so that the sensitivity to this selection is visible.

minor comments (6)

[Abstract and text] The abstract says 'three commonly used optimizer' and the body contains several typos, including 'fucntions' and 'obtianed'; please correct them.
[§4.3.1] The task label 'QM9SP_CC/DT' is a typo for 'QM9SP_CC/TZ'.
[Fig. 10 caption] The caption refers to task 'QM9GO_CASSCF', but the corresponding task in Table 1 is 'QMspinGO_CASSCF'.
[Reference 3] The reference to the Argonne Leadership Computing Facility is misspelled as 'Argone'; please fix it.
[§4.3.1] The lower-panel discussion in Figure 9 contains a repeated sentence; one of the two occurrences should be removed.
[Fig. 8] The figure caption uses 'FLOPS' while the text and Table 2 refer to 'FLOP count'; please use consistent terminology.

Circularity Check

1 steps flagged · score 2.0 of 10

Mild test-set model selection in the scheduling evaluation; otherwise the learnability results are genuine out-of-sample predictions.

other [Section 4.3.1 (Application: Optimal Scheduling), first paragraph; MAE definition in Section 3.1]
"For the scheduling optimization for all seven tasks ..., the QML model with the best representation (lowest MAE with maximum number of training points) was used which in all cases was FCHL."

The MAE used to select FCHL is the out-of-sample test error defined in Section 3.1: 'the test errors, measured as mean absolute errors (MAE), were calculated as a function of training set size.' The same test labels therefore enter twice: once to choose the representation that will be used in the scheduling simulation, and once to report the prediction accuracy that motivates the resulting 10-90% overhead reductions. This is a mild evaluation leak rather than a definitional collapse: the wall-time predictions are not constructed from the training labels, and the scheduling overhead is computed from real run times, so the central learnability result retains independent content.

full rationale

The paper's central claim—that computational cost is learnable across chemical space—rests on kernel ridge regression models trained on measured wall/CPU/FLOP timings and evaluated on disjoint test molecules. Section 3.1 states that hyperparameters 'were optimized only within the training set using random sub-sampling cross validation,' and learning curves are reported as a function of training-set size, so the prediction errors are genuine out-of-sample errors rather than refits of the training labels. The scheduling application is also not circular in its main mechanism: Section 3.2.2 uses out-of-sample QML timing estimates as input to a first-fit-decreasing bin-packing simulator, and overhead is computed from the real run times in the data, so the 10-90% reduction is a simulation result conditioned on those predictions. No load-bearing self-citation was found: FCHL and QMLcode are cited as tools or representations, not as a uniqueness proof or as the source of the scheduling result. The one mild circular element is in Section 4.3.1, where the representation used for scheduling is selected as the one with the 'lowest MAE'—and MAE is the test-set error defined in Section 3.1. Thus the test labels are used twice: once to choose FCHL over BoB and once to report the prediction quality behind the scheduling improvement. This is an evaluation-protocol leak rather than a definitional reduction; it does not make the predicted wall times equal to the inputs by construction, and it could be fixed by a separate validation split. The simulator's fixed queue-delay/startup assumptions and non-preemptible-task idealization are external validity risks, not circularity.

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

The central claim rests on kernel regression assumptions, a smoothing assumption about computational cost, post-hoc exclusion thresholds, and fixed scheduler model parameters. No new physical entities are introduced; treating wall time as a molecular quasi-property is a framing choice, not an entity.

free parameters (4)

KRR kernel width sigma per task and representation = sigma_BoB 51.2 to 204.8; sigma_FCHL 12.8 to 409.6 (SI Table 1)
Selected by random sub-sampling cross-validation inside the training set for each of the seven tasks; the reported prediction errors depend on these widths.
KRR regularization lambda per task and representation = 1e-9 to 1e-5 (SI Table 1)
Cross-validated together with sigma; small values are typical for near-noise-free data, but they still control the regression fit.
I/O-overhead exclusion thresholds per task = 3%, 5%, 10%, 30%, 50% for the five Molpro tasks
Runs above the threshold are removed before training as heavy-I/O outliers; thresholds vary by task and were chosen by inspection of Figure 3.
Scheduler startup and queue parameters = 30 seconds job start, 1 hour queue time
Chosen from observed queue statistics at sciCORE and CSCS; the overhead reduction numbers depend on these fixed simulation values.

assumptions (5)

domain assumption KRR interpolation is valid when kernel similarity between molecular representations tracks the target property.
The whole method assumes wall time is learnable from BoB or FCHL representations; Section 3.1 frames learning as interpolation between training points.
domain assumption Computational cost is a reasonably smooth function of molecular structure for each task.
Stated in the Introduction as the computational cost of a single point should be reasonably smooth over chemical space, and invoked to justify QML; the paper itself notes saddle-point discontinuities in the toy system.
ad hoc to paper High wall-to-CPU ratios are purely I/O noise and can be excluded without biasing the timing distribution.
Section 3.1 excludes runs above per-task thresholds; no analysis shows that excluded runs are not part of the true scheduling workload.
ad hoc to paper The scheduling simulator faithfully represents production schedulers with idempotent uninterruptible tasks and fixed startup and queue overhead.
Section 3.2.2 states the idempotent and uninterruptible assumption and the 30 second and 1 hour values; no real scheduler deployment validates the simulation.
domain assumption perf FLOP counts measure the computational cost of PNO-LCCSD(T)-F12 single points.
Section 2.4 and 4.2.5 use FLOPs as a clean cost measure; perf measures retired FLOPs on the specific node, which may not capture memory-bandwidth-bound phases.

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Cite this review

Pith. "Pith review of Machine learning the computational cost of quantum chemistry." pith.science (2026). https://pith.science/paper/EP672L5Z

@misc{pith2026190806714,
  author       = {Pith},
  title        = {Pith review of: Machine learning the computational cost of quantum chemistry},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EP672L5Z}},
  note         = {Machine review of arXiv:1908.06714}
}

read the original abstract

Computational quantum mechanics based molecular and materials design campaigns consume increasingly more high-performance compute resources, making improved job scheduling efficiency desirable in order to reduce carbon footprint or wasteful spending. We introduce quantum machine learning (QML) models of the computational cost of common quantum chemistry tasks. For 2D non-linear toy systems, single point, geometry optimization, and transition state calculations the out of sample prediction error of QML models of wall times decays systematically with training set size. We present numerical evidence for a toy system containing two functions and three commonly used optimizer and for thousands of organic molecular systems including closed and open shell equilibrium structures, as well as transition states. Levels of electronic structure theory considered include B3LYP/def2-TZVP, MP2/6-311G(d), local CCSD(T)/VTZ-F12, CASSCF/VDZ-F12, and MRCISD+Q-F12/VDZ-F12. In comparison to conventional indiscriminate job treatment, QML based wall time predictions significantly improve job scheduling efficiency for all tasks after training on just thousands of molecules. Resulting reductions in CPU time overhead range from 10% to 90%.

Figures

Figures reproduced from arXiv: 1908.06714 by the authors.

**Figure 1.** Compute resource growth of 500 fastest public supercomputers. 40 Estimated use by chemistry and materials sciences corresponds to 35%, corresponding to 2017 usage on Swiss National Supercomputing Center. 4 on rules of medicinal chemistry, removing the bias of pre-existing building blocks in structure selection. QM9 itself is a well established benchmark data set for quantum machine learning where many different M… view at source ↗

**Figure 2.** Wall time distribution of all tasks using kernel density estimation. CSCS. We used two electronic structure codes to generate timings. Molpro 84 was used to extract both CPU and wall times for data sets i) and ii), and ORCA85 was used to extract wall times for data set iii). Further information of the data sets, the hardware, and the calculations can be found in section 3 to 4 of the SI. The retired floating point… view at source ↗

**Figure 3.** Wall to CPU time ratio (using kernel density estimation) for Molpro calculations to identify runs with high wall time overhead due to heavy I/O load on clusters. 3.2 Application: Optimal Scheduling 3.2.1 Job Array and Job Steps In many cases, efforts in computational chemistry or materials design require the evaluation of identical tasks on different molecules or materials. Distributing those tasks across a comp… view at source ↗

Figures from the paper (9 more)

**Figure 4.** Figure 4: 2D non-linear toy systems consisting of the Rosenbrock (“Rosen”) and Himmelblau (“Him”) [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]

**Figure 5.** Figure 5: Learning curves showing normalized test errors (cross validated MAE divided by median [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]

**Figure 6.** Figure 6: PCA plots of kernel elements for BoB (left) and FCHL (right) for data set QMspinSP MRCI. The weights of the two first principal components for the molecules in the data sets are plotted against each other and corresponding wall times are encoded as a heat map. Insets s…

**Figure 7.** Figure 7: Learning curves showing normalized test errors (cross validated MAE divided by median of test set) for the first two geometry optimization steps on task QMspinGO CASSCF using BoB and FCHL as representations. The model was trained on CPU times divided by the number o…

**Figure 8.** Figure 8: Learning curves showing normalized prediction errors (cross validated MAE divided by median of test set) for FLOP count and wall times on task QM9SP CC/DZ using BoB and FCHL representations. 4.3 Application: Optimal Scheduling 4.3.1 Job Array and Job Steps For the sch…

**Figure 9.** Figure 9: Scheduling efficiencies for the seven different tasks (columns) assuming a certain per [PITH_FULL_IMAGE:figures/full_fig_p016_9.png]

**Figure 10.** Figure 10: CPU time overhead and wall time for geometry optimizations compared between the simple approach and the QML approach. See text for details of the strategies. CPU time overhead given in percent relative to the bare minimum of CPU time needed. Wall time given relative t…

**Figure 3.** Figure 3: Learning curves showing normalized test errors (cross validated MAE divided by median of test set) using BoB and FCHL as representations. The model was trained on wall times. Horizontal lines correspond to the performance assuming all calculations have mean run time…

**Figure 4.** Figure 4: Learning curves showing normalized test errors (cross validated MAE divided by median of test set) using BoB and FCHL as representations. The model was trained on normalized (number of occupied orbitals to the power of 2 times number of basis functions to the power …

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

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