CRPS-LAM: Probabilistic Regional Weather Forecasting with Continuous Ranked Probability Score

Erik Larsson; Fredrik Lindsten; Joel Oskarsson; Tomas Landelius

REVIEW 4 major objections 5 minor 19 references

CRPS-LAM shows that limited-area probabilistic forecasts can be sampled in one forward pass per ensemble member, matching diffusion-model error and improving long-lead calibration.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-08-04 10:26 UTC pith:UTQGH5KR

load-bearing objection Useful regional adaptation of CRPS-based ensemble forecasting with a real speedup, but the joint-distribution claim is supported only by marginals and a heuristic. the 4 major comments →

arxiv 2510.09484 v3 pith:UTQGH5KR submitted 2025-10-10 cs.LG

CRPS-LAM: Probabilistic Regional Weather Forecasting with Continuous Ranked Probability Score

Erik Larsson , Joel Oskarsson , Tomas Landelius , Fredrik Lindsten This is my paper

classification cs.LG

keywords probabilistic weather forecastinglimited-area modelingcontinuous ranked probability scoreensemble forecastingsingle forward-pass samplingneural weather predictionhybrid CNN/GNNMEPS dataset

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper is trying to establish that probabilistic limited-area weather forecasting does not need expensive diffusion sampling: a model trained to minimize the Continuous Ranked Probability Score, with a single Gaussian noise vector injected per ensemble member, produces full forecast fields in one forward pass. On the MEPS regional dataset over northern Europe, this model—CRPS-LAM—matches the error level of a diffusion-based forecast model while sampling about 39x faster and achieving spread-skill calibration comparable to the best baseline. The result matters because it removes the main computational obstacle to operational ensemble machine-learning weather prediction at regional scale: the cost of drawing many members.

Core claim

The central claim is that matching marginal distributions per grid point and variable—summed into a fair CRPS loss—is sufficient, when combined with one shared latent noise vector and a convolutional weight-sharing backbone, to generate physically plausible regional ensemble forecasts with diffusion-level error. Ensemble members are produced by sampling a 32-dimensional Gaussian noise vector, passing it through a linear layer into conditional normalization layers, and running the CNN/GNN backbone once; the same network is rolled out autoregressively to 57 h lead times. The paper reports that this scheme yields sampling speeds on par with deterministic models, about 39x faster than the diffus

What carries the argument

The fair CRPS estimator: for each grid point and variable, the loss is the average absolute deviation between ensemble members and the observation minus half the average pairwise deviation among members, making the training objective an unbiased proper scoring rule. Stochasticity is carried by one 32-dimensional noise vector, transformed by a linear layer and injected through conditional normalization layers, so every output field in a member is a function of the same draw. The hybrid CNN/GNN backbone shares weights across spatial locations, which the paper argues promotes spatially coherent fields even though the loss itself is per-location and per-variable.

Load-bearing premise

The load-bearing premise is that conditioning every predicted value on one shared noise vector, together with convolutional weight sharing, is enough to reproduce the true joint spatial and cross-variable distribution of the atmosphere; the training loss itself only enforces correct marginal distributions at each grid point and variable.

What would settle it

Take a fixed initialization, draw 100 latent vectors, and compute a joint diagnostic that cannot factor over grid points—for example, the rank histogram of the minimum of standardized anomalies at two distant stations, or of precipitation accumulated over a catchment. If the ensemble members are pointwise well calibrated but these joint ranks are strongly non-uniform, the shared-noise mechanism has not captured the joint distribution the paper relies on; conversely, uniform joint ranks support the claim.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

Probabilistic regional forecasts can be sampled at near-deterministic cost: about 0.5 seconds per ensemble member on one A100 GPU, with members parallelizable through batched inference.
CRPS-LAM matches the RMSE and CRPS of the diffusion-based baseline on MEPS while showing less underdispersion at longer lead times, so calibration does not have to be sacrificed for speed.
The shared-noise CRPS recipe transfers from global forecasting models to the limited-area setting, with no need for per-level or per-variable loss weighting.
Autoregressive rollout to 57 h produces forecasts with fine-scale structure, retaining more high-frequency detail than a graph-based latent-variable model, though not as much as diffusion.

Where Pith is reading between the lines

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

Editorial extension: the 32-dimensional latent is the only source of ensemble diversity; varying the latent dimension or adding spatially structured noise is the clearest knob to test and likely to change performance if the joint-distribution claim holds.
Editorial extension: the paper's evidence for joint fidelity is qualitative—visual fields and energy spectra; a quantitative test such as variogram-based rank histograms or joint exceedance probabilities over connected regions would settle whether marginal calibration is enough.
Editorial extension: training uses only two autoregressive rollout steps; extending to longer rollouts is a direct next experiment, since operational forecasts beyond 57 h may expose accumulated artifacts from the shared noise vector.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

Useful regional adaptation of CRPS-based ensemble forecasting with a real speedup, but the joint-distribution claim is supported only by marginals and a heuristic.

read the letter

Colleague,

This paper does something practical: it takes the CRPS-from-marginals training idea that Alet et al. used for global models and makes it work for limited-area forecasting. The result is a probabilistic LAM that samples ensemble members in one forward pass and runs about 39× faster than Diffusion-LAM while roughly matching its error. On MEPS, RMSE and CRPS curves are close, and the energy spectra show CRPS-LAM keeps more fine-scale detail than Graph-EFM. That is a real operational gain, not a paradigm shift.

The architecture is a sensible adaptation: same backbone as Diffusion-LAM, single 32-D noise vector, conditional normalization, autoregressive rollout with fair CRPS. The appendix is honest about training instability and the risk of collapse into near-deterministic forecasts, and they report that two-step autoregressive training fixed the artifacts. That transparency counts.

Now the soft spots. The main one is exactly where the stress-test note lands: the training objective constrains univariate marginals per grid point and variable, and the paper leans on the shared noise vector plus weight sharing to claim joint fidelity. That is a reasonable heuristic, but it is not demonstrated. RMSE, CRPS, SSR, and even energy spectra are all single-field diagnostics. No metric measures whether ensemble members are mutually plausible as joint atmospheric states—e.g., consistent cloud-precipitation relationships or coherent spatial gradients. The appendix's admission that the model sometimes ignores z entirely makes this more than a theoretical worry. The final model seems fine, but "well-calibrated" is an overstatement when only marginals are checked.

Second, all headline comparisons are mean curves with no error bars or seed variation. For a paper whose main claim is "comparable to diffusion," we need some sense of variance. This is a minor fix, not a structural flaw.

Third, code is promised but not released. Given the small number of novel moving parts, reproducibility is the main barrier to checking the joint-distribution concern.

Overall, the central argument holds up: a single-forward-pass CRPS-trained LAM is competitive and fast, and the work is worth engaging with. The missing joint-distribution verification is a real gap, but it is the kind of thing that can be addressed with a proper joint metric (e.g., rank correlation fields, spectral coherence between variables) rather than a reason to reject.

Who is this for? MLWP researchers and anyone building operational ensemble nowcasting. It deserves a serious referee—send it to review, but ask for error bars, a quantitative joint-fidelity check, and the code.

Best,

Referee Report

4 major / 5 minor

Summary. The paper introduces CRPS-LAM, a probabilistic limited-area weather forecasting model. The architecture is a hybrid CNN/GNN backbone largely inherited from Diffusion-LAM, but training replaces diffusion with a fair CRPS objective evaluated per grid point and variable (Eq. 3). A single 32-dimensional Gaussian noise vector z is injected via conditional normalization layers; ensemble members are generated by sampling z and running a single forward pass. Experiments on the MEPS dataset at 10 km resolution compare CRPS-LAM to Diffusion-LAM and Graph-EFM using RMSE, CRPS, spread-skill ratio, energy spectra, and qualitative examples. The paper claims competitive deterministic skill and calibration with roughly 39x faster sampling than diffusion, while retaining fine-scale detail and physically consistent structures.

Significance. If the claims hold, the paper offers a practical speed-quality trade-off for probabilistic limited-area forecasting: it suggests that a marginal CRPS objective with a shared latent can match diffusion-model accuracy at a fraction of sampling cost. The use of the unbiased fair CRPS estimator is methodologically sound, and the autoregressive rollout evaluation procedure is standard in this literature. The reproducibility intent is positive (code and data links are provided, though code is not yet public). However, the strongest claims — that the shared latent induces a correct joint spatial/cross-variable distribution and that forecasts are 'well-calibrated' — are not supported by the evidence presented. The paper's own appendix discloses training collapse to near-deterministic forecasts, and no quantitative joint-distribution metric is reported. These gaps are load-bearing rather than cosmetic, because the value of a probabilistic LAM lies precisely in providing mutually consistent ensemble scenarios.

major comments (4)

[Section 2, Eq. (3)] The loss in Eq. (3) is a sum of per-grid-point, per-variable univariate CRPS terms; it constrains only the marginal distributions at each location and variable. The following sentence that 'all outputs depend on the same noise vector z to correctly capture the joint forecast distribution' is an assertion, not a demonstrated property. All reported diagnostics (RMSE, CRPS, SSR, per-variable energy spectra) are marginal or single-field; there is no multivariate rank histogram, no multivariate CRPS, no cross-variable correlation check, and no quantitative spatial-coherence measure for ensemble members. Since the central value of a probabilistic LAM is to provide physically consistent scenarios, this missing validation is the weakest load-bearing point. Please add a quantitative joint-distribution evaluation, or explicitly scope the claims to marginal calibration.
[Appendix B (Experiment Details)] Appendix B discloses that during training the model 'sometimes collapses into producing near-deterministic forecasts, effectively minimizing the mean absolute error of each ensemble member while neglecting the latent variable.' This directly concerns the mechanism by which the joint distribution is supposed to be generated. The manuscript does not state whether the final model exhibits residual collapse, how the issue was detected, or how often it occurred across runs. Since the method depends on z being actively used to create ensemble spread, the final model's sensitivity to z (e.g., output variance as a function of z, or effective ensemble spread) should be reported. Without this, the reader cannot tell whether the reported ensemble members are mostly deterministic outputs with small perturbations.
[Section 3, Fig. 3 and Fig. 10] The headline claims of 'comparable' errors and 'well-calibrated' forecasts are based on mean curves without error bars, multiple seeds, or significance testing. Per-variable SSR results in Fig. 10 often deviate substantially from 1 (e.g., radiation variables nlwrs_0 and nswrs_0), yet no acceptable calibration tolerance is stated. Please provide confidence intervals (e.g., bootstrap across test cases or across model seeds) and report the number of independent initializations used. Currently, the reader cannot judge whether differences between CRPS-LAM and Graph-EFM are meaningful.
[Section 3 (speed comparison)] The 'approximately 39x faster' claim is presented without specifying the number of diffusion solver steps used for Diffusion-LAM, whether inference was batched, or whether timings include data transfer and model compilation. The parenthetical 'depending on the number of solver steps' makes the number difficult to interpret. Since sampling speed is a central contribution, please provide the precise timing protocol: hardware, batch size, solver steps, and whether the comparison is per ensemble member or for a full ensemble.

minor comments (5)

[Eq. (1)-(2)] The phrase 'independent identical random variable' should be 'independent and identically distributed.' Also clarify that the double sum in Eq. (2) includes diagonal terms that vanish, and state that N >= 2 is required for the fair CRPS estimator.
[Introduction and Method] The abstract and introduction describe a 'hybrid CNN/GNN architecture,' but the method section only describes a U-Net with conditional normalization and MLP blocks. The GNN component (inherited from Larsson et al.) is not described; please specify what parts are convolutional and what parts are graph-based, or adjust the terminology.
[Figure 3 caption] The caption refers to 'normalized RMSE' and 'normalized CRPS' but does not define the normalization. State whether the normalization is per-variable standard deviation, range, or something else, and include this detail in the text or appendix.
[Section 3 / Fig. 4] For the energy spectra, clarify whether they are computed on ensemble mean fields, individual members, or both. Also note whether the 10 km subsampling cutoff is consistently applied to all models when interpreting wavenumbers above 10^2.
[Appendix B, Table 2] The training schedule lists 200 epochs at LR 1e-5 with autoregressive steps 2. Please state whether early stopping was used and how the final checkpoint was selected, especially given the reported training instability.

Circularity Check

0 steps flagged

No significant circularity: the CRPS objective is an external proper scoring rule, training and evaluation use separate MEPS data, and the joint-distribution claim is an explicitly stated assumption rather than a derived result.

full rationale

The paper's derivation chain is not circular. Eq. (3) defines the fair CRPS loss per grid point and variable, and the model is trained on MEPS training data; RMSE, CRPS, SSR and energy spectra are computed on held-out test forecasts, so the evaluation does not reduce to the training objective or to fitted values. The claimed 39x speedup is an empirical runtime comparison, not a consequence of the loss. The central probabilistic claim is explicitly hedged: 'While Eq. (3) can be minimized by only matching marginal distributions for each location and variable, similarly to [Alet et al., 2025] we rely on the fact that all outputs depend on the same noise vector z to correctly capture the joint forecast distribution.' This is an unquantified assumption about joint fidelity, not a circular reduction—it is a correctness/validation limitation, flagged here because the paper itself notes the objective only constrains marginals and Appendix B discloses near-deterministic collapse during training. Self-citations (Larsson et al. 2025 for the backbone and MEPS metrics; Oskarsson et al. 2024 for the training schedule) are used for architecture, dataset, and baselines, not to establish the paper's conclusions; the MEPS dataset is openly available and the baselines are external comparison methods. No uniqueness theorem or prior result by the same authors is invoked to force the model choice. Hence no step reduces by construction to its inputs; the only minor self-citation is the Diffusion-LAM backbone, which is not load-bearing for the paper's central contribution.

Axiom & Free-Parameter Ledger

4 free parameters · 4 axioms · 1 invented entities

The central dependency beyond standard ML training choices is the assumption that one shared noise vector, combined with CNN weight-sharing, can generate a physically coherent joint distribution even though the loss only matches pointwise marginals. The other parameters are conventional training choices or preprocessing decisions, though the unreported training ensemble size is an unexplained gap.

free parameters (4)

Latent noise dimensionality = 32
The 32-dimensional z is chosen by hand; it defines the expressiveness of the stochasticity and is not derived from data or a capacity requirement.
Training ensemble size N for fair CRPS = not reported
The fair CRPS estimator depends on N; prior work uses large early ensembles to avoid collapse. The paper does not state the N used in training, which affects the loss and the stability claims.
Autoregressive training steps = 2
Training uses up to two autoregressive steps; chosen after observing no benefit from more steps. This directly affects rollout stability at 57 h.
Spatial subsampling factor = every 4th grid point (10 km)
The MEPS native 2.5 km data is subsampled to 10 km; this determines which spectral scales are physically meaningful and influences the fine-scale-detail comparison.

axioms (4)

domain assumption MEPS analysis fields are adequate ground truth for forecast skill evaluation.
RMSE, CRPS and SSR are computed against MEPS states; if the ground truth has systematic errors, calibration claims inherit them.
ad hoc to paper A single 32-dimensional Gaussian noise vector is sufficient to represent forecast uncertainty across all grid points and variables.
Stated in Section 2: 'all outputs depend on the same noise vector z to correctly capture the joint forecast distribution'; no capacity analysis or quantitative joint-fidelity metric is provided.
domain assumption Autoregressive rollout of 3 h steps to 57 h remains stable and approximately Markovian.
Training only covers two autoregressive steps, while evaluation rolls out 19 steps; the controlled error-compounding assumption is implicit in Section 3.
standard math The fair CRPS estimator is an unbiased estimator of CRPS.
Equations (2) and (3) rely on the unbiasedness result of Ferro [2014]; this is accepted background.

invented entities (1)

Single 32-dimensional latent noise vector z no independent evidence
purpose: Injects stochasticity so that each forward pass yields an ensemble member.
There is no external falsifiable handle; its sufficiency for the joint forecast distribution is asserted in Section 2 rather than independently verified.

pith-pipeline@v1.3.0-alltime-deepseek · 13088 in / 11032 out tokens · 108998 ms · 2026-08-04T10:26:09.616303+00:00 · methodology

0 comments

read the original abstract

Limited-Area Models (LAMs) enable weather forecasting over regional domains at higher resolutions than what is computationally feasible for global models. At such high resolutions, machine learning approaches for weather prediction increasingly rely on ensemble methods to produce probabilistic forecasts. However, existing machine learning LAMs are not scalable due to relying on computationally costly diffusion models or inefficient graph neural networks. We tackle this by introducing a new hybrid CNN/GNN architecture, tailored to the LAM weather forecasting problem. Using this architecture, we construct the DET-LAM deterministic model, producing LAM forecasts both more efficiently and accurately than its graph-based competitor. We then tackle the ensemble forecasting problem, by using this architecture as a backbone for the generative model CRPS-LAM. CRPS-LAM is trained using a Continuous Ranked Probability Score (CRPS) objective, enabling efficient training and sampling in a single forward pass. This yields a speedup of $\approx \times 39$ compared to diffusion-based baselines. We evaluate our approach on regional domains in northern Europe, demonstrating that CRPS-LAM produces skillful and well-calibrated forecasts across a range of atmospheric variables.

Figures

Figures reproduced from arXiv: 2510.09484 by Erik Larsson, Fredrik Lindsten, Joel Oskarsson, Tomas Landelius.

**Figure 2.** Figure 2: Forecasts at 57 h lead time for r_2. The faded area constitutes the boundary region. Graph-EFM Diffusion-LAM CRPS-LAM Calibrated 1 2 Lead time (Days) 0.2 0.3 (a) RMSE 1 2 Lead time (Days) 0.10 0.15 (b) CRPS 1 2 Lead time (Days) 0.0 0.5 1.0 (c) SSR [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Figure 3: The mean of the normalized RMSE, CRPS, and SSR for all variables. [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

**Figure 4.** Figure 4: Energy spectra of humidity at 57 h . CRPS-LAM shows low errors, comparable with Diffusion-LAM, while retaining the fast inference speed of Graph-EFM by only requiring a single forward pass to sample the next time step. A well calibrated ensemble should have SSR ≈ 1, yet all models exhibit some degree of underdispersion. CRPS-LAM achieves ensemble calibration comparable to the bestperforming model, Graph… view at source ↗

**Figure 5.** Figure 5: An overview of the backbone model. 7 [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗

**Figure 6.** Figure 6: A description of each U-Net block. The training procedure for CRPS-LAM is summarized in [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗

**Figure 7.** Figure 7: An ensemble forecasts for each variable at 57 h. [PITH_FULL_IMAGE:figures/full_fig_p025_7.png] view at source ↗

**Figure 8.** Figure 8: The RMSE results for each variable. 26 [PITH_FULL_IMAGE:figures/full_fig_p026_8.png] view at source ↗

**Figure 9.** Figure 9: The CRPS results for each variable. 27 [PITH_FULL_IMAGE:figures/full_fig_p027_9.png] view at source ↗

**Figure 10.** Figure 10: The SSR results for each variable. 28 [PITH_FULL_IMAGE:figures/full_fig_p028_10.png] view at source ↗

**Figure 11.** Figure 11: The energy spectra for each variable at the lead times 3 h, 30 h, and 57 h. [PITH_FULL_IMAGE:figures/full_fig_p029_11.png] view at source ↗

**Figure 12.** Figure 12: The energy spectra for each variable at the lead times 3 h, 30 h, and 57 h. [PITH_FULL_IMAGE:figures/full_fig_p030_12.png] view at source ↗

**Figure 13.** Figure 13: The energy spectra for each variable at the lead times 3 h, 30 h, and 57 h. [PITH_FULL_IMAGE:figures/full_fig_p031_13.png] view at source ↗

**Figure 14.** Figure 14: The energy spectra for each variable at the lead times 3 h, 30 h, and 57 h. [PITH_FULL_IMAGE:figures/full_fig_p032_14.png] view at source ↗

discussion (0)

Reference graph

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