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arxiv: 2606.20165 · v1 · pith:QJHKNGAMnew · submitted 2026-06-18 · ⚛️ physics.ao-ph

PRecover 1.0: Process Rate Recovery with Machine Learning

Miriam Simm , Tom Beucler , Corinna Hoose This is my paper

Pith reviewed 2026-06-26 15:01 UTC · model grok-4.3

classification ⚛️ physics.ao-ph
keywords machine learningmicrophysical process ratesICON modelcloud microphysicspost-processingrandom forestsneural networksprocess rate recovery
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The pith

Machine learning recovers most microphysical process rates from standard ICON outputs for short accumulation intervals.

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

The paper introduces PRecover, a post-processing method that trains random forests, gradient boosting models, and neural networks to reconstruct microphysical process rates missing from numerical weather model outputs. It demonstrates that these rates can be recovered from standard cloud variables in high-resolution European ICON simulations, with a two-step classification-regression approach for warm-rain and ice processes. Recovery succeeds for most rates accumulated over 10 minutes or less but grows harder for longer intervals, while direct physics recalculation works for some rates like accretion but fails for autoconversion and heterogeneous ice nucleation. The method also supplies calibrated uncertainty intervals and shows the models transfer to unseen regional domains. This addresses storage limits that prevent saving all process rates during simulations, enabling better study of precipitation formation and aerosol-cloud interactions.

Core claim

PRecover recovers most of the process rates that are accumulated over output time steps of 10 minutes or less from standard cloud variables using supervised learning, but the values are increasingly difficult to recover for rates accumulated over longer accumulation intervals. A physics-based baseline shows direct recalculation succeeds for rates such as accretion and self-collection but not for autoconversion, rain melting or heterogeneous ice nucleation. The approach includes calibrated prediction intervals via conformalized quantile regression and demonstrates spatial transferability across different regional domains and simulation settings.

What carries the argument

PRecover, a data-driven post-processing approach that trains random forests, gradient boosting models, and feed-forward neural networks on cloud variables to recover microphysical process rates via a two-step classification-regression approach.

If this is right

  • Direct recalculation from stored variables succeeds for accretion and self-collection but not for autoconversion, rain melting or heterogeneous ice nucleation.
  • Recovery accuracy holds for accumulation intervals of 10 minutes or less and declines for longer intervals.
  • The trained models transfer to different regional domains and simulation settings not used in training.
  • Calibrated prediction intervals provide quantified uncertainty for each recovered rate.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Simulations could store fewer variables during runtime and still support detailed post-hoc process analysis.
  • The method could extend to other microphysics schemes or global models if similar input variables are available.
  • Longer accumulation intervals might require additional input features or hybrid physics-ML constraints to maintain accuracy.
  • Output frequency choices in model design could be guided by which processes prove hardest to recover.

Load-bearing premise

Standard cloud variables saved in ICON output contain enough information to reconstruct the missing process rates via supervised learning on high-resolution European simulations.

What would settle it

Applying the trained models to ICON simulations with accumulation intervals longer than 10 minutes and finding that recovery errors exceed those reported for short intervals, or observing failure on a new domain outside the training regions.

Figures

Figures reproduced from arXiv: 2606.20165 by Corinna Hoose, Miriam Simm, Tom Beucler.

Figure 1
Figure 1. Figure 1: Overview of PRecover. Left: ICON advances the dynamical core, fast-physics package, and model output with t dyn = t fast/5, t fast, and t out, respectively. At each fast-physics step, it computes the microphysical process rates ∂tq X k and ∂tn X k , stored alongside standard ICON variables to construct the training, validation, and test datasets. Right: PRecover predicts instantaneous and accumulated proce… view at source ↗
Figure 2
Figure 2. Figure 2: 24 hour-accumulated precipitation amount on 23 June 2023 from RADOLAN (Bartels, 2004) on the ICON-D2 domain. Here, we focus on a heavy snowfall event that occurred on 7 and 8 March 2018 (Ko et al., 2022, [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Simulation domain and 24 hour-accumulated precipitation amount on 7 March 2018 over the Korean Peninsula for the ICE-POP campaign [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Simulation domain and sea ice fraction on 3 September 2020 over the Arctic for the MOSAiC campaign. 13 [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Classification performance (measured by F1 score and MCC) is high for most instantaneous and accumulated process rates but decreases with increasing output time steps; except of heterogeneous ice nucleation (QI_HET), where low scores slightly increase with increasing output time step. Shown are the F1 score (hatched) and the MCC (solid) for the logistic regression baseline models (orange), the RFs (pink), … view at source ↗
Figure 6
Figure 6. Figure 6: Regression performance (measured by R 2 score) is moderate to high in most cases but varies across process rates and model architectures and generally decreases with increasing output time step. Shown is the R 2 score for the linear regression baseline models (orange), the RFs (pink), gradient boosting models (green) and NNs (blue). A missing bar indicates R 2 < 0. The full results are given in Table A2. B… view at source ↗
Figure 7
Figure 7. Figure 7: The MCC of the classification models remains stable up to an output time step of 10 minutes for autoconversion, accretion, and rain evaporation, but decreases at longer accumulation intervals. Heterogeneous ice nucleation is the main exception, with MCC increasing toward longer output time steps. Shown are MCC values for the logistic regression baseline models (orange), RFs (pink), gradient boosting model … view at source ↗
Figure 8
Figure 8. Figure 8: The R 2 of the regression models remains high up to an output time step of 2 minutes for autoconversion and 10 minutes for accretion and rain evaporation, but decreases at longer accumulation intervals. Heterogeneous ice nucleation follows the same trend with generally lower scores, except for the instantaneous case. Shown are R 2 scores for the linear regression baseline models (orange), RFs (pink), gradi… view at source ↗
Figure 9
Figure 9. Figure 9: The combined classification-regression models show moderate to high predictive performance, which varies across process rates and is slightly lower for the accumulated process rates. Heterogeneous ice nucleation is the clear exception, with R 2 values close to zero for both instantaneous and the accumulated cases, indicating no predictive performance. Shown is the R 2 score (solid bars) and R 2 log score (… view at source ↗
Figure 10
Figure 10. Figure 10: The distributions of the true (dark blue) and predicted (orange) values of the instantaneous microphysical process rates with a 10-minute output time step show general agreement, but the frequency of very small and very large values is slightly underestimated. The predicted distribution of heterogeneous ice nucleation misses the second peak at high true values. Shown are distributions of the true and pred… view at source ↗
Figure 11
Figure 11. Figure 11: The distributions of the true (dark blue) and predicted (orange) values of the microphysical process rates accumulated over a 10-minute output time step show general agreement, but the frequency of very small and very large values is underestimated. The predicted distribution of heterogeneous ice nucleation severely underestimates the second peak at high true values. Shown are distributions of the true an… view at source ↗
Figure 12
Figure 12. Figure 12: The combined classification-regression models show moderate to high predictive performance (measured by R 2 score and R 2 log), which varies across process rates and is slightly lower for the accumulated process rates. Shown is the R 2 score (solid bars) and R 2 log score (hatched bars) for the 10-minute output time step for instantaneous process rates (dark blue colors) and accumulated process rates (dar… view at source ↗
Figure 13
Figure 13. Figure 13: The prediction intervals for the combined classification-regression models achieve high empirical coverage with PICP values close to the nominal 90% level. The rounding operation (Eq. (11)) increases the empirical coverage to above 90% for most process rates. This is except for heterogeneous ice nucleation (QI_HET) for which PICP ≈ 45% across different output time steps. In this case, split conformal pred… view at source ↗
Figure 14
Figure 14. Figure 14: The normalized mean prediction interval width (NMPIW) generally increases for longer output time steps, reflecting increasing predictive uncertainty. The improved empirical coverage ( [PITH_FULL_IMAGE:figures/full_fig_p031_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: The vertical profiles of the spatio-temporal mean microphysical process rates show overall agreement between the ML predictions and the ICON model output, except for heterogeneous ice nucleation (QI_HET). For autoconversion (QR_AC) and rain freezing (QR_RF), the predicted mean values are substantially lower than the ICON output across height levels. Shown are vertical profiles of eight process rates accum… view at source ↗
Figure 16
Figure 16. Figure 16: The spatial distributions of autoconversion (QR_AC) and accretion (QR_ACC), accumulated over a 10-minute output time step for 23 June 2023 with the ICON-D2 domain, show similar patterns between the ICON model output (left) and the ML prediction (center) with small to moderate regional differences, normalized to the respective range of the process rate (right). The spatial distributions are averaged over a… view at source ↗
Figure 17
Figure 17. Figure 17: The vertical profiles of the spatio-temporal mean microphysical process rates show overall agreement between the ML predictions and the ICON model output, except for heterogeneous ice nucleation (QI_HET). For autoconversion (QR_AC) the predicted mean values are substantially lower than the ICON output across height levels. Shown are vertical profiles of eight process rates accumulated over a 10-minute out… view at source ↗
Figure 18
Figure 18. Figure 18: The spatial distributions of autoconversion (QR_AC) and accretion (QR_ACC), accumulated over a 10-minute output time step for 7 March 2018 with the ICE-POP domain ( [PITH_FULL_IMAGE:figures/full_fig_p034_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: The vertical profiles of the spatio-temporal mean microphysical process rates show overall agreement between the ML predictions and the ICON model output. For heterogeneous ice nucleation (QI_HET) the predicted mean values are substantially lower than the ICON output across height levels. Shown are vertical profiles of eight process rates accumulated over a 10-minute output time step for 3 September 2020 … view at source ↗
Figure 20
Figure 20. Figure 20: The spatial distributions of rain freezing (QR_RF) and the total riming rate (QX_RIMING), accumulated over a 10-minute output time step for 3 September 2020 with the MOSAiC domain ( [PITH_FULL_IMAGE:figures/full_fig_p036_20.png] view at source ↗
read the original abstract

Comprehensive information on cloud microphysical process rates from numerical simulations allows for better understanding of precipitation formation pathways and aerosol-cloud interactions. However, resource limitations often make it impractical to include all microphysical process rates in the model output, limiting in-depth analyses. To address this shortcoming, we introduce PRecover, a data-driven post-processing approach to recover microphysical process rates that are not stored during runtime from standard output of a numerical weather prediction model. In particular, we train random forests, gradient boosting models, and feed-forward neural networks to recover microphysical process rates from a two-moment bulk microphysics scheme in the ICOsahedral Nonhydrostatic (ICON) model. We use cloud variables as input, obtained from high-resolution simulations in a limited-area setup over Europe. Warm-rain and ice microphysical process rates are recovered with a two-step classification-regression approach for both instantaneous and accumulated process rates. As a physics-based baseline, we assess whether process rates can be directly recalculated from stored ICON output variables. Accurate recalculation is possible for process rates such as accretion and self-collection but not for the autoconversion, rain melting or heterogeneous ice nucleation rate. Using PRecover, we successfully recover most of the process rates that are accumulated over output time steps of 10 minutes or less, but the values are increasingly difficult to recover for rates accumulated over longer accumulation intervals. To quantify predictive uncertainty, we provide calibrated prediction intervals through conformalized quantile regression. We demonstrate spatial transferability of the models with two case studies over different regional domains and simulation settings unseen during training.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The paper introduces PRecover 1.0, a supervised ML post-processing method (random forests, gradient boosting, feed-forward NNs) to recover warm-rain and ice microphysical process rates from standard ICON cloud variables. It employs a two-step classification-regression approach for both instantaneous and accumulated rates, compares against a physics-based recalculation baseline, supplies calibrated prediction intervals via conformalized quantile regression, and tests spatial transferability on held-out regional domains. The central result is that most rates are recoverable for accumulation intervals of 10 min or less, with recovery becoming increasingly difficult at longer intervals.

Significance. If the quantitative results hold, the work provides a practical route to extract additional microphysical information from existing NWP output without reruns or expanded storage, directly supporting analyses of precipitation pathways and aerosol-cloud interactions. Explicit strengths include the physics baseline that isolates under-determined rates, the conformal prediction for uncertainty, and the spatial transferability checks on unseen domains.

major comments (2)
  1. [Results] Results section: the central claim that 'most' process rates are successfully recovered for accumulation intervals ≤10 min is not supported by any reported quantitative metrics (R², MAE, bias, or error distributions) for individual rates or intervals; without these the extent of success cannot be evaluated and the trend with accumulation interval remains qualitative.
  2. [Physics baseline] Physics baseline subsection: the statement that accurate recalculation is possible for accretion and self-collection but not for autoconversion, rain melting or heterogeneous ice nucleation is presented without the explicit formulas, stored-variable mappings, or error statistics used to reach that conclusion, leaving the baseline comparison load-bearing for the claim that ML adds value.
minor comments (3)
  1. [Abstract] Abstract: replace the qualitative phrase 'successfully recover most' with at least one concrete performance number (e.g., median R² across rates for the 10-min case) so readers can immediately gauge the result.
  2. [Methods] Methods: list the exact set of ICON cloud variables used as input features and any normalization or feature-selection steps applied before training.
  3. [Figures] Figure captions: ensure every panel reports the accumulation interval and the specific process rate shown so that the degradation trend with interval is immediately visible.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive report and positive recommendation. We address each major comment below.

read point-by-point responses

  1. Referee: [Results] Results section: the central claim that 'most' process rates are successfully recovered for accumulation intervals ≤10 min is not supported by any reported quantitative metrics (R², MAE, bias, or error distributions) for individual rates or intervals; without these the extent of success cannot be evaluated and the trend with accumulation interval remains qualitative.

    Authors: We agree that quantitative metrics are required to substantiate the claim. The revised manuscript will add tables and supplementary figures reporting R², MAE, bias, and full error distributions for every individual process rate at each accumulation interval (1 min, 5 min, 10 min, 30 min, 60 min). These will also quantify the degradation trend with longer intervals. revision: yes

  2. Referee: [Physics baseline] Physics baseline subsection: the statement that accurate recalculation is possible for accretion and self-collection but not for autoconversion, rain melting or heterogeneous ice nucleation is presented without the explicit formulas, stored-variable mappings, or error statistics used to reach that conclusion, leaving the baseline comparison load-bearing for the claim that ML adds value.

    Authors: We accept that the physics baseline description is insufficiently detailed. The revision will include the explicit diagnostic formulas, the precise mappings from stored ICON two-moment variables to each rate, and the quantitative error statistics (mean relative error, RMSE) that support the distinction between accurately recalculable rates and under-determined ones. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper applies supervised ML (random forests, gradient boosting, neural nets) to map standard ICON cloud variables to microphysical process rates, training and validating on independent high-resolution simulations where the target rates are directly available. The central claim is an empirical test of information content in stored fields, supported by physics baselines, held-out regional domains, and conformal prediction intervals. No derivation, equation, or claim reduces to its own inputs by construction; no self-citation chains, ansatzes, or fitted-input renamings appear in the load-bearing steps.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the sufficiency of saved cloud variables as inputs and on the representativeness of the European high-resolution training simulations for the target rates.

free parameters (1)
  • ML model hyperparameters
    Random forests, gradient boosting, and neural network architectures and training settings are tuned to the simulation data.
axioms (1)
  • domain assumption Saved cloud variables contain sufficient information to reconstruct the target process rates
    The two-step classification-regression pipeline uses only standard output variables as inputs.

pith-pipeline@v0.9.1-grok · 5818 in / 1271 out tokens · 29741 ms · 2026-06-26T15:01:06.382199+00:00 · methodology

discussion (0)

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