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REVIEW 3 major objections 4 minor 17 cited by

AIFS-CRPS: Ensemble forecasting using a model trained with a loss function based on the Continuous Ranked Probability Score

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Training a weather model with a proper score produces a stochastic ensemble that beats a 9-km physics-based system for most fields and lead times.

desk verdict Medium-range skill claims look credible; the subseasonal/MJO comparison is confounded by mismatched reforecast years and should not be taken at face value. read the letter →

arxiv 2412.15832 v1 pith:UE2F3S5H submitted 2024-12-20 physics.ao-ph

Simon Lang , Mihai Alexe , Mariana C. A. Clare , Christopher Roberts , Rilwan Adewoyin , Zied Ben Bouallègue , Matthew Chantry , Jesper Dramsch
show 10 more authors
Peter D. Dueben Sara Hahner Pedro Maciel Ana Prieto-Nemesio Cathal O'Brien Florian Pinault Jan Polster Baudouin Raoult Steffen Tietsche Martin Leutbecher
This is my paper · ORCID
classification physics.ao-ph
keywords ensembleforecastingmachinelearningweatherpredictioncontinuousrankedprobabilityscoreproperscoringrulestochasticneuralnetworksubseasonalforecastMadden-JulianOscillationAIFS
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 introduces AIFS-CRPS, a machine-learned ensemble weather model trained end-to-end to minimize the almost fair Continuous Ranked Probability Score (afCRPS). The central claim is that a single proper scoring objective, applied to a small training ensemble, is enough to teach a stochastic neural network to generate a well-calibrated, arbitrarily large ensemble of exchangeable forecast members. The authors report that this model scores better than the physics-based IFS ensemble for the majority of medium-range upper-air and surface variables, and that it stays competitive into the subseasonal range, with lower biases and better Madden-Julian Oscillation forecasts than the operational subseasonal system. If correct, this means the computationally heavy perturbed physics ensemble can be replaced by one cheap stochastic model evaluation per member.

What carries the argument

The central object is the almost fair CRPS, defined as $\text{afCRPS}_\alpha = \alpha\,\text{fCRPS} + (1-\alpha)\,\text{CRPS}$, with $\epsilon = (1-\alpha)/M$, interpolating between the standard CRPS and the fair CRPS; it is rearranged as a double sum of non-negative terms so it remains safe under reduced precision. The second engine is noise-conditioned architecture: independent Gaussian noise per member is embedded and injected via conditional layer normalizations, making ensemble members exchangeable and the model stochastic at inference. Reference-field truncation, $x_{t+1} = U(D(x_t)) + f(x_t)$ with downsampling operator $D$ and upsampling operator $U$, prevents the accumulation of small-scale artifacts during autoregressive rollout.

What would settle it

Take the same trained AIFS-CRPS and issue forecasts from (a) unperturbed analyses, (b) operational IFS perturbed starts, and (c) starts with the revised perturbation amplitude mentioned in Section 6; if spread-error ratios and CRPS move materially across these, then the reported ensemble skill depends on perturbations the model was never trained on. Alternatively, compute afCRPS in float16 for a case with M-1 members exactly at the observation to see whether the degeneracy it avoids actually returns.

Watch

Extended reading notes

Core claim

AIFS-CRPS is a variant of the AIFS transformer-based architecture in which all standard layer normalizations in the processor are replaced by conditional layer normalizations driven by random Gaussian noise. A small ensemble is propagated in parallel during training and scored against the deterministic ERA5 analysis with afCRPS, a convex combination of the fair CRPS and the standard CRPS that removes most of the finite-ensemble-size bias while avoiding the degeneracy that afflicts the fair score in low precision. In inference the model is rolled out autoregressively from each perturbed IFS initial condition, with each member using its own random seed. The paper's central discovery is that this simple training objective produces a stochastic model whose members keep realistic small-scale variability at long lead times and whose ensemble skill exceeds the 9-km IFS ensemble for most variables and lead times in the medium range, with strong subseasonal performance, particularly for the Madden-Julian Oscillation.

Load-bearing premise

The model is trained on a small ensemble initialized from a single deterministic analysis, but evaluated from perturbed IFS initial conditions it never saw during training; the paper notes that revised perturbation amplitudes change reliability, so the unseen perturbation statistics are load-bearing for the reported calibration and skill.

Editorial extensions

If this is right

  • If the reported skill holds, a single afCRPS objective suffices to generate a full ensemble, without singular-vector perturbations or stochastic physics schemes.
  • Inference cost is one model evaluation per member per 6-hour step, so a 15-day forecast for one member takes about one minute at O96 and four minutes at N320 on an A100 GPU, making large ensembles cheap to produce.
  • Subseasonal predictions at two to six weeks are competitive with or better than the operational IFS subseasonal system, especially for the Madden-Julian Oscillation, despite training only on short-range rollout steps.
  • Raising input resolution from O96 to N320 improves most surface variables, suggesting the same training recipe can be pushed to higher resolutions and additional variables.
  • The system still depends on physics-based analyses for training data and initialization, so it refines rather than replaces the current data-assimilation infrastructure.

Reading between the lines

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

  • Extension: the hyperparameter $\alpha$ in afCRPS offers a tunable knob between bias correction and stability; one could expect the optimal $\alpha$ to vary with ensemble size, precision, and variable, a recipe transferable to other probabilistic machine-learning forecasters.
  • Extension: the paper notes that first tests with a revised initial perturbation amplitude improve reliability, which suggests a natural next step is to train with perturbed initial conditions so the model learns initial-condition uncertainty directly instead of aliasing it into model uncertainty.
  • Extension: the surrogate MJO index used here omits outgoing longwave radiation, yet still yields strong skill; evaluating a full RMM index with OLR would test whether the missing radiative channel matters for MJO prediction in this model.
  • Extension: since 46-day forecasts remain stable despite training on rolls of at most 72 hours, long-range skill appears to emerge from short-range training, hinting that similar objectives could support seasonal prediction with modest changes.
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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

3 major / 4 minor

Summary. The paper presents AIFS-CRPS, a machine-learned global ensemble weather prediction model based on the AIFS architecture, trained by minimizing a newly introduced 'almost fair' CRPS loss (afCRPS) that blends the fair CRPS with the standard CRPS to avoid degeneracy in finite-precision training. The model injects Gaussian noise through conditional layer normalization to generate exchangeable ensemble members. The authors evaluate 15-day medium-range forecasts from two resolutions (O96 and N320) against the operational IFS ensemble, and 46-day subseasonal forecasts against IFS reforecasts. They report that AIFS-CRPS outperforms IFS for most upper-air and surface variables in the medium range, maintains realistic spectral variability without smoothing, and shows lower biases and improved MJO forecast skill at subseasonal lead times, while acknowledging remaining over-dispersion and stratospheric degradation.

Significance. If the results hold, AIFS-CRPS is a significant advance: it demonstrates that a single CRPS-style scoring-rule objective can yield a stochastic ML ensemble that rivals a high-resolution physics-based ensemble at a fraction of inference cost, with stable small-scale variability and useful subseasonal forecasts. The afCRPS formulation is a useful methodological contribution to scoring-rule-based training, as it addresses a real degeneracy of the fair CRPS while remaining simple to implement. The evaluation is unusually thorough for a preprint, including analysis-based and observation-based verification, significance testing in the scorecards, and explicit discussion of limitations such as stratospheric degradation and over-dispersion. The main caveat is that the subseasonal/MJO comparison rests on mismatched reforecast periods and perturbation methods, so those specific claims are not yet established; the medium-range comparison is much cleaner and supports the central skill claim more strongly.

major comments (3)
  1. [Section 5.3, Figures 10–11] The subseasonal and MJO comparisons are confounded by two asymmetries. The AIFS-CRPS reforecasts are initialized once per week over 2018–2022, while the IFS reforecasts are from 2023 only; matching calendar dates across years does not control for the different MJO states and verification targets sampled. In addition, the AIFS-CRPS subseasonal reforecasts use ERA5-EDA perturbations, whereas the IFS reforecasts use ERA5-EDA plus singular-vector perturbations, so the two systems differ in initial-condition perturbation methodology as well. The block-bootstrap significance testing and lead-time-dependent climatologies do not remove these confounds because the verification data themselves differ between the two samples. The conclusion in Section 7 that AIFS-CRPS exhibits 'increased Madden-Julian Oscillation (MJO) forecast skill compared to ECMWF's operational subseasonal forecasting system' therefore needs either a matched-period reforecast comparison or a substantially tempered claim.
  2. [Sections 2.3 and 6] The model is trained by propagating a small ensemble from the same deterministic ERA5 analysis, but all reported medium-range results are obtained by initializing from perturbed IFS operational initial conditions. Section 6 states that 'First tests with a revised initial perturbation amplitude show improved reliability (not shown),' which indicates that the ensemble calibration, and hence the spread and CRPS results in Figures 5–9, may depend on the perturbation amplitude used at inference. Since that amplitude is an externally chosen parameter not seen in training, the paper should either report the sensitivity of the headline scores to the initial perturbation amplitude or explicitly mark the reliability results as conditional on the current operational perturbation scheme.
  3. [Section 5.2, Figures 6–7] The medium-range skill comparison is based on a single eight-month period (1 February to 30 September 2024). While the scorecards include significance testing, the conclusion in Section 7 is stated without seasonal qualification: 'AIFS-CRPS forecast skill is higher than that of the 9 km physics-based IFS medium-range ensemble for most upper-air fields and surface variables.' This period does not include a Northern Hemisphere winter, so the claim should be restricted to the evaluated season or supported by additional seasons before being stated as a general result.
minor comments (4)
  1. [Sections 2.1 and 2.2] Two different equations are both numbered (1): the reference-field truncation update in Section 2.1 and the CRPS definition in Section 2.2. Please renumber to avoid ambiguity.
  2. [Section 2.1 and Table 1] Section 2.1 states that the O32 grid has approximately 2.5° resolution, while Table 1 lists O32 as approximately 3.0° resolution; these values should be made consistent.
  3. [Figure 5 caption] The Figure 5 caption contains the typo 'nothern extra-tropics'; it should read 'northern extra-tropics'.
  4. [Section 2.2] The choice of α = 0.95 in the afCRPS loss is not accompanied by any sensitivity analysis; a brief study of α values near 1 would help justify the 'almost fair' approximation and show that the results are not sensitive to this hyperparameter.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the afCRPS objective is defined independently of the verification data, and the headline skill claims are evaluated on out-of-sample dates against an independent physics-based ensemble.

full rationale

The derivation chain is not circular. The afCRPS loss (Eq. 3) is a convex combination of the standard CRPS and the fair CRPS, with a pre-chosen hyperparameter alpha=0.95; it is not defined in terms of the verification scores or fitted to the IFS benchmark. The medium-range skill claims are based on forecasts from 1 February to 30 September 2024, after the 1979-2017 training and 2016-2023 fine-tuning windows, and are verified against operational IFS analyses, radiosonde and SYNOP observations, so the comparison is out-of-sample and externally benchmarked. The subseasonal evaluation relies on self-cited methodology (Roberts et al. 2023; Roberts and Leutbecher 2024) for climatologies and significance testing, but these are general verification methods with stated assumptions that do not assume AIFS-CRPS skill, so they are independent support rather than load-bearing circularity. Two non-circular caveats are worth noting separately: the subseasonal AIFS-CRPS reforecasts (2018-2022) are compared with IFS reforecasts from 2023 only, which is a sampling confound, and the fine-tuning window includes 2018-2022, so the label 'out-of-sample' for the subseasonal evaluation is not strictly accurate; neither issue makes any prediction equivalent to its training input by construction.

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

The central claims rest on standard ML weather prediction assumptions (analysis as truth, noise injection sufficiency) and on the new afCRPS loss whose parameters are chosen by hand. No new physical entities are introduced.

free parameters (5)
  • alpha (afCRPS mixing parameter) = 0.95
    Chosen as a hyperparameter in Equation (3). Alpha close to 1 to approximate fair CRPS while avoiding degeneracy. Not fit to data.
  • Minimum pressure scaling factor = 0.2 (O96 only)
    Introduced in section 2.2 to cap the pressure-dependent loss weighting. Not used for N320. Affects stratospheric performance.
  • Reference field truncation resolution = O32
    Chosen in section 2.1 for downsampling/upsampling of the reference state. A design choice affecting spectral behavior.
  • Training ensemble size = 4 (O96), 2 (N320)
    Section 4; chosen to balance cost. Affects the afCRPS estimate quality during training.
  • Noise channel count = unspecified
    Section 2; the number of noise channels n is stated to equal processor grid points times noise channels, but the channel count is not given.
assumptions (4)
  • domain assumption Optimizing the afCRPS with a small training ensemble yields a model whose ensemble members are exchangeable samples from the forecast distribution.
    The paper assumes that minimizing a proper score with a finite training ensemble produces a well-calibrated stochastic model; section 2.2.
  • domain assumption ERA5 and the operational IFS analysis are appropriate ground truth for training and verification.
    Standard practice in ML weather prediction; sections 3 and 5. The model is trained and verified against the same analysis system, which may bias comparisons.
  • ad hoc to paper Gaussian noise injected via conditional layer normalization is sufficient to represent forecast uncertainty.
    The architecture choice in section 2; not derived from first principles. The paper evaluates it empirically.
  • domain assumption The model trained on 6-hourly steps and rollouts up to 72 hours will generalize to 15-day and 46-day forecasts.
    The paper relies on autoregressive stability; section 5 evaluates this empirically.

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Cite this review

Pith. "Pith review of AIFS-CRPS: Ensemble forecasting using a model trained with a loss function based on the Continuous Ranked Probability Score." pith.science (2026). https://pith.science/paper/UE2F3S5H

@misc{pith2026241215832,
  author       = {Pith},
  title        = {Pith review of: AIFS-CRPS: Ensemble forecasting using a model trained with a loss function based on the Continuous Ranked Probability Score},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UE2F3S5H}},
  note         = {Machine review of arXiv:2412.15832}
}
read the original abstract

Over the last three decades, ensemble forecasts have become an integral part of forecasting the weather. They provide users with more complete information than single forecasts as they permit to estimate the probability of weather events by representing the sources of uncertainties and accounting for the day-to-day variability of error growth in the atmosphere. This paper presents a novel approach to obtain a weather forecast model for ensemble forecasting with machine-learning. AIFS-CRPS is a variant of the Artificial Intelligence Forecasting System (AIFS) developed at ECMWF. Its loss function is based on a proper score, the Continuous Ranked Probability Score (CRPS). For the loss, the almost fair CRPS is introduced because it approximately removes the bias in the score due to finite ensemble size yet avoids a degeneracy of the fair CRPS. The trained model is stochastic and can generate as many exchangeable members as desired and computationally feasible in inference. For medium-range forecasts AIFS-CRPS outperforms the physics-based Integrated Forecasting System (IFS) ensemble for the majority of variables and lead times. For subseasonal forecasts, AIFS-CRPS outperforms the IFS ensemble before calibration and is competitive with the IFS ensemble when forecasts are evaluated as anomalies to remove the influence of model biases.

Figures

Figures reproduced from arXiv: 2412.15832 by the authors.

Figure 1
Figure 1. Probabilistic training of AIFS-CRPS. A small ensemble of atmospheric states is propagated forward in time [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. 24-hr (left column) and 240-hr (right column) forecasts of meridional wind at 850 hPa, from perturbed [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Geopotential at 500 hPa of a 300-hour forecast from perturbed member 1 of the AIFS-CRPS ensemble when [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: Spectra of geopotential at 500 hPa (a, b) and temperature at 850 hPa (c, d) for different lead times. Step 0 h [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: AIFS-CRPS N320 (blue, solid line) and IFS ensemble (green, dashed line) CRPS of 2 m temperature for [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 9
Figure 9. Figure 9: Ensemble mean RMSE (solid line) and ensemble spread (dotted line) for geopotential at 500 hPa (a, b and c) [PITH_FULL_IMAGE:figures/full_fig_p011_9.png]
Figure 10
Figure 10. Figure 10: Score cards summarizing differences between AIFS-CRPS and IFS in the fair continuous ranked probability [PITH_FULL_IMAGE:figures/full_fig_p014_10.png]
Figure 11
Figure 11. Figure 11: (a) Bivariate correlations for an MJO index calculated from 200 hPa and 850 hPa zonal wind anomalies for [PITH_FULL_IMAGE:figures/full_fig_p015_11.png]
Figure 12
Figure 12. Figure 12: Hovmöller diagrams showing the evolution of zonal wind anomalies at 850 hPa meridionally averaged from [PITH_FULL_IMAGE:figures/full_fig_p015_12.png]
Figure 13
Figure 13. Figure 13: Phase diagrams based on the surrogate Real-time Multivariate MJO index described in the text for 46-day [PITH_FULL_IMAGE:figures/full_fig_p016_13.png]

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Reviewed August 11, 2026 · model on record in the stance chip above.