Quantum Neural Networks for Cloud Cover Parameterizations in Climate Models

Arthur Grundner; Lorenzo Pastori; Mierk Schwabe; Veronika Eyring

REVIEW 2 major objections 5 minor 125 references

Quantum Neural Networks for Cloud Cover Parameterizations in Climate Models

T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper shows that small data re-uploading quantum circuits can learn cloud cover from coarse-grained climate simulation data with accuracy comparable to classical neural networks, and that both outperform a standard semi-empirical…

desk verdict A careful, honest QML-vs-classical benchmark on real climate data; the parity claim holds, but the 'outperforms standard parameterizations' claim is only tested against a re-fitted simplified Xu-Randall scheme. read the letter →

arxiv 2502.10131 v1 pith:RJLOKJM4 submitted 2025-02-14 quant-ph physics.ao-ph

Lorenzo Pastori , Arthur Grundner , Veronika Eyring , Mierk Schwabe This is my paper

classification quant-phphysics.ao-ph

keywords quantummachinelearningneuralnetworkscloudcoverparameterizationclimatemodelsdatare-uploadingFisherinformationmatrixshotnoiseDYAMOND

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 tests whether quantum neural networks (QNNs) can serve as data-driven cloud cover parameterizations in climate models. It trains two small parameterized quantum circuits, with roughly 200 trainable parameters, on coarse-grained cloud cover data from high-resolution ICON simulations and compares them with classical neural networks of the same size. The reported result is that the QNNs are comparable to the classical networks in prediction accuracy, generalization with training-set size, and training dynamics, and that both outperform a simplified conventional cloud scheme, the Xu-Randall scheme. The study also shows that training and inference remain stable under finite measurement noise when enough measurement shots are used, and that variance regularization can lower the needed shot count. If the result holds, quantum machine learning is a workable ansatz for learning subgrid cloud processes, although the comparison is made entirely offline.

What carries the argument

The central object is the data re-uploading quantum circuit: input features are uploaded multiple times as single-qubit rotation angles, interleaved with variational blocks containing entangling gates, and finally read out as a trainable weighted average of $\hat{\sigma}_z$ expectation values plus a bias, $f_\theta(x) = b + \sum_n w_n \langle \hat{\sigma}^z_n\rangle_{\theta}(x)$. The two main ansatzes, labelled XYZ and ZZXY, use six or eight qubits and roughly 200 trainable parameters. Training minimizes mean squared error with the Adam optimizer, using gradients from the parameter-shift rule. The supporting machinery is the Fisher information matrix and its normalized effective dimension for trainability analysis, and a variance-regularized loss $\mathrm{MSE} + \lambda\,\mathrm{MPV}$ that reduces the number of measurement shots needed.

What would settle it

A decisive test would be to couple each trained QNN and the matched classical NN into the ICON model and run multi-week online simulations: if the QNN's offline parity disappears under coupling, or if the coupled QNN produces larger cloud radiative biases or unstable climate statistics than the classical NN, the claim that QNNs are comparable parameterizations would collapse.

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Extended reading notes

Core claim

The paper's central claim is that parameterized quantum circuits with data re-uploading can learn the mapping from coarse-grained atmospheric state variables (specific humidity, cloud water, cloud ice, temperature, pressure, wind speed, height, latitude) to cloud cover as accurately as classical feed-forward neural networks of matched parameter count, and that both outperform a simplified Xu-Randall scheme. In the noiseless simulations, the QNNs' $R^2$ is roughly $0.01$ lower than the classical networks' but their vertical mean cloud cover profile, spatial bias maps, and the scaling of test error with $N_{\mathrm{train}}$ are essentially the same. The paper also claims that the QNNs have a flatter Fisher information spectrum and a higher normalized effective dimension, yet this geometrical advantage does not translate into faster or more stable training on this task. Under shot noise, training is stable with about $10^4$ measurement shots, and variance regularization with $\lambda=0.005$ stabilizes training already at $10^2$ shots.

Load-bearing premise

The load-bearing premise is that the coarse-grained DYAMOND data, after removing all cells with zero condensate and applying the fitted input and output transformations, faithfully represents the cloud-cover relationship a climate model needs, so that offline accuracy on a holdout from the same simulation windows is a meaningful measure of a deployable parameterization.

Editorial extensions

If this is right

Small QNNs with about 200 trainable parameters can learn cloud cover to nearly the accuracy of classical NNs of the same size.
Both QNNs and classical NNs beat the simplified Xu-Randall scheme in offline accuracy, strengthening the case for data-driven parameterizations.
Test error follows the same roughly $1/\sqrt{N_{\mathrm{train}}}$ scaling for quantum and classical networks until saturation near the parameter count.
QNN training and prediction remain stable under shot noise when expectation values are estimated with about $10^4$ shots, and variance regularization brings the needed shot count down to about $10^2$.
A flatter Fisher spectrum and higher effective dimension for QNNs do not, in this task, produce faster or more stable training than classical NNs.

Reading between the lines

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

If the parity is real, the practical case for QNN parameterizations must rest on quantum-specific advantages such as expressivity per parameter or hardware integration, because the paper finds no accuracy advantage.
The offline evaluation is a favorable test: removing all zero-condensate cells and fitting the transforms to the training window makes the regression easier, so online coupled runs could shrink the margin over the Xu-Randall baseline or expose instabilities.
A natural next experiment is to apply the same matched architectures to harder parameterization targets such as radiation or convection, where the paper itself notes a quantum-classical separation might appear.
The variance-regularization result suggests a concrete shot-budget recipe for near-term devices, but wall-clock time and hardware noise are not assessed, so that bridge remains open.

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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

Colleague,

The genuinely new piece is the matched-parameter-count comparison between QNNs and classical NNs on a coarse-grained cloud cover regression task from the DYAMOND ICON simulations. Nothing in the cited literature does that for cloud cover. The main result—QNNs land within about 0.01 R² of matched classical NNs, with both scaling the same way in Ntrain and trainability—is directly supported by the experiments, including the ensemble spreads. The shot-noise analysis is also real work: the variance formula is correct, and the variance-regularization result is a credible demonstration that the QNNs train stably at nshots around 10^4, with regularization helping at 100 shots.

The paper is honest where it hurts. The authors report that classical NNs are consistently slightly better, and they explicitly say the FIM spectrum and effective dimension do not translate into faster or more stable training. That negative result is a useful counterweight to Abbas et al., and they handled it without hand-waving.

The load-bearing weakness is the abstract claim that both ansatzes outperform standard parameterizations used in climate models. The only conventional baseline is a simplified two-parameter Xu-Randall scheme with α and β fit by MSE minimization on the training set. That is the best possible version of a deliberately simplified scheme; beating it does not establish beating the operational cloud cover scheme in ICON or any other climate model. The parity claim does not depend on this baseline and survives. The stress-test note is right: this part of the central claim is unsupported.

Two minor soft spots. The dataset removes all zero-condensate cells, so the task is cloud cover conditional on condensate presence, not the full prediction a deployed parameterization faces; the paper says this, but the abstract does not. The R² gap of about 0.01 is reported without error bars, though the ensemble spreads in Fig. 3 give a rough sense. Code and data are available only on request, which weakens reproducibility.

Who should read it: anyone working on QML benchmarks or ML parameterizations. It won't change your picture of quantum advantage, but it is a clean case study of how to run a fair QNN/NN comparison. It deserves a serious referee. I would ask the authors to put the 'outperforms standard parameterizations' claim in proportion, or add a real operational baseline, before acceptance.

Recommendation: send it to review; conditional on revision of the baseline claim.

Referee Report

2 major / 5 minor

Summary. The paper investigates whether quantum neural networks (QNNs) can serve as cloud cover parameterizations for climate models, comparing QNNs to classical neural networks (NNs) on a coarse-grained DYAMOND/ICON dataset. The authors design two QNN architectures (XYZ and ZZXY) with data re-uploading, train them using the parameter-shift rule, and compare performance, generalization, and trainability against classical NNs with matched parameter counts. They also study the effect of finite measurement shots and show that variance regularization can stabilize training with few shots. The main positive results are that QNNs achieve accuracy comparable to classical NNs, that both outperform a simplified Xu-Randall baseline on the tested conditional cloud-cover task, and that no clear correlation is found between FIM geometry and training dynamics.

Significance. If the parity claim holds, the paper provides a careful, well-documented empirical demonstration that small QNNs can learn meaningful patterns in climate data, with an extensive comparison across architectures, training instances, and noise levels. The paper includes parameter-shift gradients, a detailed FIM derivation in Appendix E, and reproducible-looking experimental protocols with 20 or more training instances per configuration. These are strengths. However, the broader claim that QNNs and NNs 'outperform standard parameterizations used in climate models' is not supported by the evidence presented, which restricts the significance of the result until the baseline comparison is broadened or the claim is appropriately scoped.

major comments (2)

[Section 4.1, Eq. (7)] The abstract and introduction state that both ansatzes outperform 'standard parameterizations used in climate models,' but the only baseline evaluated is Eq. (7), a simplified Xu-Randall scheme whose two constants α and β are fitted to the training set. Beating this two-parameter diagnostic, which the paper itself calls a simplified version, does not establish outperformance of operational schemes such as the ICON cloud cover parameterization or other published cloud-cover schemes. The parity claim (QNN ≈ NN) is independent of this baseline and is supported, but the 'outperforming standard parameterizations' claim is load-bearing and currently unsupported. Please either replace the baseline with an operational scheme (or a published, non-fitted reference) or rephrase the claim to refer specifically to the fitted Xu-Randall diagnostic on this dataset.
[Section 2, data filtering] The paper removes all cells with zero cloud condensate before training and evaluation, so the task is cloud cover conditional on the presence of condensate, not the full parameterization problem that a deployed scheme faces (which must also predict zero cloud cover for condensate-free cells). The abstract and conclusions describe the result as developing cloud cover parameterizations generally, which overstates the scope of the experiments. This data choice should be stated as an explicit limitation in the abstract or conclusions, or the models should be evaluated on the full dataset to demonstrate the unconditional performance.

minor comments (5)

[Eq. (4)] Equation (4) has an unbalanced parenthesis: it reads '(fθ(xi) − yi)' with an extra closing parenthesis before the square; it should be '(fθ(xi) − yi)' or '(fθ(xi) − yi)^2' with the parenthesis placed correctly.
[Section 4.1, text after Fig. 3] The sentence 'Similar conclusions can be drawn the the global bias maps' contains a duplicated 'the'; it should read 'drawn from the global bias maps'.
[Table 1 caption] The caption contains 'T able 1' with a space, which appears to be a formatting artifact.
[References] Several references have formatting issues: 'Eyring et al., 2021, 2024,?' contains a literal question mark, and the reference to 'Monta˜nez-Barrera' has an accent rendering artifact.
[Section 5.2] The variance regularization section uses a fixed λ=0.005 and acknowledges that Kreplin and Roth propose a dynamical schedule; it would be useful to state whether the results are sensitive to the chosen λ, since the noiseless analysis shows a trade-off between MSE and MPV.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the QNN/NN parity comparison is an independent empirical benchmark; the fitted Xu-Randall baseline creates an overstatement concern rather than a circular step.

full rationale

This is an empirical benchmarking paper rather than a derivation, and its central claims are supported by held-out evaluation. The QNN prediction f_theta(x) is defined by Eq. (3) and trained by minimizing Eq. (4) on the DYAMOND training set; its reported MSE and R2 are computed on a non-overlapping test set in Section 4.1, so the parity claim versus classical NNs is not constructed from the same data used for fitting. The input and output transforms in Appendix B are fitted to the training set but are applied identically to the QNN and the classical NN, so they cannot force a QNN-classical parity result. The paper's self-citations, such as Grundner et al. (2022, 2024), justify variable selection and coarse-graining but are not used as a uniqueness theorem or as a substitute for the benchmark comparison. The only feature even resembling a fitted-input issue is the Xu-Randall baseline in Eq. (7), whose constants alpha and beta are MSE-fitted on the training set; however, the paper explicitly labels this a 'simplified version' of Xu-Randall, and beating this baseline does not make the QNN outputs equal to any fitted constant, nor is the outperform claim manufactured by the baseline construction. The gap between the abstract's phrase 'outperforming standard parameterizations used in climate models' and the actually tested simplified, fitted baseline is a correctness or support concern, not a circularity. Overall, no load-bearing circular step was identified; the low score reflects only minor non-load-bearing self-citations and an overbroad baseline wording in the abstract.

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

The central claims rest on five domain or modeling assumptions and on transforms fitted to the data, not on any new physical postulate. The fitted constants (input and output transform parameters, Xu-Randall coefficients) are applied symmetrically to all compared models, so they do not by themselves explain the reported parity, but they do shape the task on which parity is claimed. No new particles, forces, mediators, dimensions, or conserved quantities are introduced; the QNN ansatze are standard constructions cited to prior literature.

free parameters (5)

Output transform parameters a, b, c of g(x) = a=1.29407913, b=-3.20011015, c=0.70308237
Appendix B: chosen so that transformed cloud cover targets are approximately uniformly distributed in [0,1]; fitted to the training target distribution before training either model.
Input transform parameters (b, xlow, xhigh) for qv, qc, qi, hw = qv: b=0.25, xlow=1e-7, xhigh=0.025; qc: b=0.25, xlow=0, xhigh=0.00145; qi: b=0.25, xlow=0, xhigh=0.00055; hw: b=0.5…
Appendix B: parameters of h(x) chosen to flatten the sharply peaked input histograms and retain tail variability; fitted to the training set statistics, then applied identically to QNN and NN inputs.
Xu-Randall baseline coefficients alpha and beta = alpha=4.034e4, beta=0.9942
Section 4.1: optimized via MSE minimization over the training set, so the 'standard scheme' baseline is an optimally fitted two-parameter form, not the published Xu-Randall coefficients.
Variance regularization weight lambda = lambda=0.005
Section 5.2 and Fig. 7(a,b): lambda chosen from a scan of values; the paper notes a dynamical schedule could further improve results.
Hyperparameters (learning rate, batch size, epochs, nenc, nvar) = Adam lr=0.001, batch=100, 150-200 epochs; nenc and nvar per architecture in Table 1
Appendix C and Section 3.3: hand-chosen and stated as approximately optimal for both ansatzes; held fixed across the QNN versus NN comparison, so they affect absolute performance but not the relative comparison.

assumptions (5)

domain assumption Coarse-grained DYAMOND ICON output provides a valid supervised target for cloud cover parameterization (binary cloudiness at the 1e-6 kg/kg condensate threshold, coarse-grained to an 80 km R2B5 grid).
Section 2 and Fig. 1; inherited from Giorgetta et al. (2022) and the authors' prior parameterization papers (Grundner et al. 2022, 2024). If the high-resolution simulation target is biased, every compared model inherits the same bias, so parity is unaffected but the 'outperforming standard schemes' claim would rest on a biased target.
domain assumption Dropping all zero-condensate cells does not distort the learning task for a deployed parameterization.
Section 2, final paragraph. Disclosed and physically motivated (no cloud cover without condensate), but it truncates the target distribution; the trained models are never evaluated on the full cell population a parameterization would face.
ad hoc to paper Equal trainable parameter count is the appropriate complexity match for the QNN versus classical NN comparison.
Section 4.1 and Table 1. Parameter counts are matched only approximately (200 or 201 versus 203; 109 or 114 versus 119), and the classical architectures were selected by an architecture search (Appendix C) while QNN architectures were selected from the tested ansatze; this is a reasonable but not unique fairness criterion.
standard math The Fisher information matrix for these deterministic regression models is computed under a Gaussian output density with fictitious variance.
Appendix E, Eqs. (E1)-(E4): the derivation sets p_theta(y|x) to a normal centered on f_theta(x); this follows Pennington and Worah (2018) and Karakida et al. (2020) and is internally consistent, but the effective dimension values depend on this convention.
domain assumption Random cell-level sampling yields train and test sets independent enough for the reported accuracy comparison.
Sections 4.1-4.2: test cells are sampled at random space-time locations disjoint from training cells, but atmospheric fields are strongly correlated in space and time, so near-duplicate leakage between train and test cells is possible; this could inflate absolute accuracy for all models equally.

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Cite this review

Pith. "Pith review of Quantum Neural Networks for Cloud Cover Parameterizations in Climate Models." pith.science (2026). https://pith.science/paper/RJLOKJM4

@misc{pith2026250210131,
  author       = {Pith},
  title        = {Pith review of: Quantum Neural Networks for Cloud Cover Parameterizations in Climate Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RJLOKJM4}},
  note         = {Machine review of arXiv:2502.10131}
}

read the original abstract

Long-term climate projections require running global Earth system models on timescales of hundreds of years and have relatively coarse resolution (from 40 to 160 km in the horizontal) due to their high computational costs. Unresolved subgrid-scale processes, such as clouds, are described in a semi-empirical manner by so called parameterizations, which are a major source of uncertainty in climate projections. Machine learning models trained on short high-resolution climate simulations are promising candidates to replace conventional parameterizations. In this work, we explore the potential of quantum machine learning (QML), and in particular quantum neural networks (QNNs), to develop cloud cover parameterizations. QNNs differ from their classical counterparts, and their potentially high expressivity turns them into promising tools for accurate data-driven schemes to be used in climate models. We perform an extensive comparative analysis between several QNNs and classical neural networks (NNs), by training both ansatzes on data coming from high-resolution simulations with the ICOsahedral Non-hydrostatic weather and climate model (ICON). Our results show that the overall performance of the investigated QNNs is comparable to that of classical NNs of similar size, i.e., with the same number of trainable parameters, with both ansatzes outperforming standard parameterizations used in climate models. Our study includes an analysis of the generalization ability of the models as well as the geometrical properties of their optimization landscape. We also investigate the effects of finite sampling noise, and show that the training and the predictions of the QNNs are stable even in this noisy setting. These results demonstrate the applicability of QML to learn meaningful patterns in climate data, and are thus relevant for a broad range of problems within the climate modeling community.

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write newline

" write newline "" before.all 'output.state := FUNCTION string.to.integer 't := t text.length 'k := #1 'char.num := t char.num #1 substring 's := s is.num s "." = or char.num k = not and char.num #1 + 'char.num := while char.num #1 - 'char.num := t #1 char.num substring FUNCTI...

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

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