基于适当评分规则的扩散模型用于概率天气预报

# Proper Scoring Rule-based Diffusionfor Probabilistic Weather Forecasting
Source: [https://arxiv.org/html/2609.38632](https://arxiv.org/html/2609.38632)
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Giung NamAffiliation:KAISTHyungi LeeAffiliation:Kookmin UniversityKyunghyun ChoAffiliation:New York UniversityByoungwoo ParkAffiliation:KAISTJuho LeeEmail:[clearclouds@kaist\.ac\.kr](mailto:[email protected])Email:[bw\.park@kaist\.ac\.kr](mailto:[email protected])Email:[juholee@kaist\.ac\.kr](mailto:[email protected])Corresponding author:, ,Affiliation:KAIST

###### Abstract

Recent probabilistic weather forecasters train stochastic predictors with the continuous ranked probability score \(CRPS\) to generate each ensemble member in a single forward pass\. These models learn the predictive distribution from the forecast context alone, which becomes difficult at longer forecast horizons where uncertainty is high\. To learn the predictive distribution more effectively, we introduce auxiliary conditional denoising tasks that predict the same future state from the context and its corrupted version, which provides partial future information that can reduce prediction ambiguity\. Building on distributional diffusion models, we learn the conditional distributions of these tasks with a single stochastic predictor by minimizing a proper scoring rule across noise levels\. At inference, the predictor can still generate each ensemble member in a single forward pass at the fully corrupted endpoint\. Standard CRPS training is recovered as the endpoint\-only special case of our formulation, so our framework extends existing CRPS\-based forecasters with only additional conditioning inputs\. Controlled experiments show that the auxiliary tasks improve one\-step forecasting across architectures, with larger gains at longer forecast horizons\. The gains extend to high\-dimensional global weather forecasting under both training from scratch and fine\-tuning, along with improved calibration and potential benefits for generalization under distribution shift\.

## 1Introduction

Numerical weather prediction \(NWP\) has long underpinned weather forecasting by simulating atmospheric evolution\. Ensembles of these simulations represent uncertainty from imperfect initial conditions and model approximations\([Leutbecher and Palmer, 2008](https://arxiv.org/html/2609.38632#bib.bib15);[Slingo and Palmer, 2011](https://arxiv.org/html/2609.38632#bib.bib13)\), but are computationally demanding at high resolution\([Bauer et al\., 2015](https://arxiv.org/html/2609.38632#bib.bib10)\)\. Recently, deterministic machine learning \(ML\) models have achieved competitive forecast skill at substantially lower inference cost than NWP\([Bi et al\., 2023](https://arxiv.org/html/2609.38632#bib.bib9);[Lam et al\., 2023](https://arxiv.org/html/2609.38632#bib.bib5)\), with ensemble forecasts generated from perturbed initial conditions\([Kurth et al\., 2023](https://arxiv.org/html/2609.38632#bib.bib25);[Chen et al\., 2023](https://arxiv.org/html/2609.38632#bib.bib23)\)\. However, deterministic models trained to predict the conditional mean can produce overly smooth ensemble members\([Price et al\., 2025](https://arxiv.org/html/2609.38632#bib.bib8)\)\. This limitation has motivated growing interest in probabilistic ML models for weather forecasting\.

In probabilistic weather forecasting, models learn predictive distributions over future atmospheric states conditioned on the forecast context\([Gneiting and Katzfuss, 2014](https://arxiv.org/html/2609.38632#bib.bib14)\)\. Diffusion\-based forecasters generate spatially detailed ensemble members from noise through iterative conditional denoising\([Price et al\., 2025](https://arxiv.org/html/2609.38632#bib.bib8);[Couairon et al\., 2026](https://arxiv.org/html/2609.38632#bib.bib6)\)\. Yet this iterative sampling requires repeated model evaluations for each member at every forecast step, making ensemble forecasting costly\. Recent CRPS\-based forecasters offer an efficient alternative, training stochastic predictors with objectives based on the continuous ranked probability score \(CRPS\) to generate each ensemble member in a single forward pass per forecast step\([Alet et al\., 2025](https://arxiv.org/html/2609.38632#bib.bib41);[Bonev et al\., 2025](https://arxiv.org/html/2609.38632#bib.bib40);[Lang et al\., 2026](https://arxiv.org/html/2609.38632#bib.bib11)\)\. Such one\-step ensemble generation is therefore promising for probabilistic weather forecasting\.

However, these forecasters learn the predictive distribution from the forecast context alone, and such context\-only training can become harder as forecast uncertainty grows\. When forecasting several days ahead in one step, for example, we observe that CRPS\-trained models tend to overfit \([Section3\.1](https://arxiv.org/html/2609.38632#S3.SS1)\)\. To learn the predictive distribution more effectively, we draw on evidence that jointly learning related tasks can enhance performance on the target task\([Caruana, 1997](https://arxiv.org/html/2609.38632#bib.bib22);[Maurer et al\., 2016](https://arxiv.org/html/2609.38632#bib.bib3)\)and consider auxiliary prediction tasks while retaining one\-step forecasting\. This leads to the following question\.

![Refer to caption](https://arxiv.org/html/2609.38632v1/procast_intro_editable_v23.png)Figure 1:Comparison of probabilistic weather forecasting methods\. Diffusion\-based models \(left\) train a deterministic denoiser, while CRPS\-based models \(middle\) train a stochastic predictor to generate ensembles\. Our method \(right\) retains the stochastic predictor and adds a sampled corrupted future and its noise level as conditioning inputs \(purple\)\. These inputs introduce auxiliary conditional denoising tasks learned through shared parameters, while preserving one\-step ensemble generation\.How can auxiliary tasks help one\-step forecasters learn the predictive distribution under high uncertainty?

We address this question with auxiliary tasks that predict the same future from the context and an auxiliary variable carrying partial information about the future\. We hypothesize that learning these tasks can help a single forecaster better capture the context\-future relationship, as this information reduces prediction ambiguity\. To include the original forecasting problem, the auxiliary variable should admit an uninformative case and, ideally, allow control over how much future information it provides\. Diffusion forward process\([Sohl\-Dickstein et al\., 2015](https://arxiv.org/html/2609.38632#bib.bib39)\)naturally satisfies these requirements: the corrupted future becomes uninformative at the fully corrupted endpoint, and the noise level controls the amount of future information\([Kingma et al\., 2021](https://arxiv.org/html/2609.38632#bib.bib38)\)\. We therefore construct auxiliary conditional denoising tasks that predict the same clean future from its corrupted version and the forecast context\.

To learn such conditional denoising tasks while retaining one\-step forecasting, we introduce a proper scoring rule\-based diffusion framework built on distributional diffusion models\([De Bortoli et al\., 2025](https://arxiv.org/html/2609.38632#bib.bib28), DDMs;\)\. In our framework, a stochastic predictor learns the conditional distributions of these tasks and generates ensemble members directly at the fully corrupted endpoint\. With component\-wise CRPS as the scoring rule, we show that CRPS training is an endpoint\-only special case of our formulation\. As illustrated in[Figure1](https://arxiv.org/html/2609.38632#S1.F1), our framework can therefore extend existing CRPS\-based weather forecasters by adding only the corrupted future and its noise level as conditioning inputs\.

Through extensive experiments across diverse architectures and forecast horizons, we show that our formulation improves one\-step forecasting over standard CRPS training, with larger gains under high uncertainty\. These gains extend to high\-dimensional global weather forecasting under both training from scratch and fine\-tuning\. We also observe improved generalization under distribution shift and in tropical cyclone cases, consistent with a possible regularizing effect of auxiliary tasks\.

Our contributions are summarized below\.

- •We introduce a proper scoring rule\-based diffusion framework that uses auxiliary conditional denoising tasks to learn predictive distributions more effectively under high uncertainty\.
- •We revisit standard CRPS training as the endpoint\-only special case of our formulation and generalize it with the auxiliary conditional denoising tasks\.
- •Building on this connection, we analyze how auxiliary conditional denoising tasks can improve one\-step forecasting and support our interpretation through extensive controlled experiments\.
- •We find that these benefits extend to high\-dimensional weather forecasting across architectures and training regimes, with only minor conditioning changes to existing CRPS\-based forecasters\.

## 2Preliminaries

In this section, we briefly review probabilistic weather forecasting, focusing on scoring rule\-based and diffusion\-based approaches\. Further background including related work is provided in[AppendixA](https://arxiv.org/html/2609.38632#A1)\.

##### Probabilistic Weather Forecasting

LetXτX^\{\\tau\}denote the atmospheric state at time stepτ\\tau, with consecutive steps separated by a fixed forecast interval\. Probabilistic weather forecasting commonly adopts an autoregressive formulation, factorizing the distribution over forecast trajectories into one\-step conditional distributions\([Price et al\., 2025](https://arxiv.org/html/2609.38632#bib.bib8);[Alet et al\., 2025](https://arxiv.org/html/2609.38632#bib.bib41)\),

p\(X1:T∣c0\)=∏τ=0T−1p\(Xτ\+1∣cτ\),p\\left\(X^\{1:T\}\\mid c^\{0\}\\right\)=\\textstyle\{\\prod\_\{\\tau=0\}^\{T\-1\}\}p\\left\(X^\{\\tau\+1\}\\mid c^\{\\tau\}\\right\),\(1\)where the forecast contextcτc^\{\\tau\}consists of the current stateXτX^\{\\tau\}and may additionally include previous states and static and forcing variables\. For notational simplicity, we writeX=Xτ\+1X=X^\{\\tau\+1\}andc=cτc=c^\{\\tau\}, so the goal is to model the predictive distributionp⁡\(X∣c\)p\(X\\mid c\)\([Gneiting and Katzfuss, 2014](https://arxiv.org/html/2609.38632#bib.bib14)\)\. In practice, a learned modelpθ​\(X∣c\)p\_\{\\theta\}\(X\\mid c\)generates an ensemble ofMMfuture states\{X^\(m\)\}m=1M\\smash\{\\\{\\hat\{X\}^\{\(m\)\}\\\}\_\{m=1\}^\{M\}\}approximating this distribution, and each member is rolled out by updating the context with its predicted state\.

##### Scoring Rules for Weather Forecasting

Scoring rules assign a scalar loss to a predictive distribution and an observed outcome\. Throughout this work, we adopt the convention that lower scores are better\. They are*proper*if expected loss is minimized when the predictive distribution matches the true distribution, and*strictly proper*if the minimizer is unique\([Gneiting and Raftery, 2007](https://arxiv.org/html/2609.38632#bib.bib12)\)\.

In weather forecasting, a common choice is CRPS\([Matheson and Winkler, 1976](https://arxiv.org/html/2609.38632#bib.bib17)\), which is given by

SCRPS\(pθ\(⋅∣c\),X\)=𝔼\[∥X^−X∥1\]−12𝔼\[∥X^−X^′∥1\],S\_\{\\texttt\{CRPS\}\}\\left\(p\_\{\\theta\}\(\\cdot\\mid c\),X\\right\)=\\mathbb\{E\}\\left\[\\\|\\hat\{X\}\-X\\\|\_\{1\}\\right\]\-\\frac\{1\}\{2\}\\mathbb\{E\}\\left\[\\\|\\hat\{X\}\-\\hat\{X\}^\{\\prime\}\\\|\_\{1\}\\right\],\(2\)whereX^,X^′∼i\.i\.d\.pθ\(⋅∣c\)\\hat\{X\},\\hat\{X\}^\{\\prime\}\\overset\{\\mathrm\{i\.i\.d\.\}\}\{\\sim\}p\_\{\\theta\}\(\\cdot\\mid c\)and∥⋅∥1\\\|\\cdot\\\|\_\{1\}sums absolute differences over forecast channels and grid points\. WithMMensemble members, we estimateSCRPSS\_\{\\texttt\{CRPS\}\}with the unbiasedfair CRPS\([Ferro, 2014](https://arxiv.org/html/2609.38632#bib.bib16)\),

S^CRPS​\(\{X^\(m\)\}m=1M,X\)=1M​∑m=1M‖X^\(m\)−X‖1−12​M​\(M−1\)​∑m≠n‖X^\(m\)−X^\(n\)‖1\.\\hat\{S\}\_\{\\texttt\{CRPS\}\}\\left\(\\\{\\hat\{X\}^\{\(m\)\}\\\}\_\{m=1\}^\{M\},X\\right\)=\\frac\{1\}\{M\}\\sum\_\{m=1\}^\{M\}\\\|\\hat\{X\}^\{\(m\)\}\-X\\\|\_\{1\}\-\\frac\{1\}\{2M\(M\-1\)\}\\sum\_\{m\\neq n\}\\\|\\hat\{X\}^\{\(m\)\}\-\\hat\{X\}^\{\(n\)\}\\\|\_\{1\}\.\(3\)Beyond evaluation, CRPS can also serve as a training objective for learning predictive distributions\. Weather forecasting models can directly learnpθ\(⋅∣c\)p\_\{\\theta\}\(\\cdot\\mid c\)by minimizing a CRPS\-based ensemble loss and generate each ensemble member in a single forward pass\([Alet et al\., 2025](https://arxiv.org/html/2609.38632#bib.bib41);[Lang et al\., 2026](https://arxiv.org/html/2609.38632#bib.bib11)\)\.

##### Diffusion\-based Weather Forecasting

Diffusion\-based forecasters model the predictive distributionp⁡\(X∣c\)p\(X\\mid c\)by iteratively denoising random noiseε\\varepsiloninto future states conditioned on the context\.

Following[Lipman et al\. \(2023\)](https://arxiv.org/html/2609.38632#bib.bib35), we consider the forward corruption using a linear noise schedule,

qt​\(Xt∣X\)=𝒩⁡\(Xt∣\(1−t\)​X,t2​I\),Xt=\(1−t\)​X\+t​ε,ε∼𝒩⁡\(0,I\),q\_\{t\}\(X\_\{t\}\\mid X\)=\\mathcal\{N\}\\bigl\(X\_\{t\}\\mid\(1\-t\)X,t^\{2\}I\\bigr\),\\qquad X\_\{t\}=\(1\-t\)X\+t\\varepsilon,\\quad\\varepsilon\\sim\\mathcal\{N\}\(0,I\),\(4\)wheret∈\[0,1\]t\\in\[0,1\]is the noise level\. Reverse sampling uses a denoiserDθ​\(c,Xt,t\)D\_\{\\theta\}\(c,X\_\{t\},t\)trained to minimize

ℒDiffusion​\(θ\)=∫01wt​𝔼c,X,Xt∼qt\(⋅∣X\)​\[‖X−Dθ​\(c,Xt,t\)‖2\]​𝑑t,\\mathcal\{L\}\_\{\\mathrm\{Diffusion\}\}\(\\theta\)=\\int\_\{0\}^\{1\}w\_\{t\}\\,\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}c,X,X\_\{t\}\\sim q\_\{t\}\(\\cdot\\mid X\)\\end\{subarray\}\}\\left\[\\left\\\|X\-D\_\{\\theta\}\(c,X\_\{t\},t\)\\right\\\|^\{2\}\\right\]\\,\\mathrm\{d\}t,\(5\)wherewt≥0w\_\{t\}\\geq 0weights noise levels\. The optimal denoiser outputs the posterior mean,D∗\(c,Xt,t\)=𝔼\[X∣c,Xt,t\]\{D^\{\*\}\(c,X\_\{t\},t\)=\\mathbb\{E\}\[X\\mid c,X\_\{t\},t\]\}\. Within DDIM sampling\([Song et al\., 2021](https://arxiv.org/html/2609.38632#bib.bib36)\), the denoiser output serves as a point\-estimate approximation of the clean\-state posterior,p\(⋅∣c,Xt,t\)≈δDθ​\(c,Xt,t\)p\(\\cdot\\mid c,X\_\{t\},t\)\\approx\\delta\_\{D\_\{\\theta\}\(c,X\_\{t\},t\)\}, and this approximation can degrade sample quality under coarse time discretization\([De Bortoli et al\., 2025](https://arxiv.org/html/2609.38632#bib.bib28)\)\. Generating high\-quality ensemble members therefore typically requires fine time discretization, with repeated denoiser evaluations for each member incurring substantial inference cost\([Price et al\., 2025](https://arxiv.org/html/2609.38632#bib.bib8)\)\.

## 3Proper Scoring Rule\-based Diffusion Forecaster

We first examine challenges in context\-only forecasting, which learns the predictive distributionp⁡\(X∣c\)p\(X\\mid c\)of the future stateXXfrom the contextccalone\. Motivated by these challenges, we introduce auxiliary conditional denoising tasks and evaluate their benefits\. We then build on distributional diffusion models\([De Bortoli et al\., 2025](https://arxiv.org/html/2609.38632#bib.bib28), DDMs;\)to formalize a proper scoring rule\-based diffusion forecaster that retains one\-step ensemble generation\. Finally, we show that standard CRPS training can be viewed as an endpoint\-only special case of our formulation and present the resulting algorithm\.

### 3\.1Motivating Observations: Challenges of Context\-Only Forecasting

Figure 2:FCN3\([Bonev et al\., 2025](https://arxiv.org/html/2609.38632#bib.bib40)\)train and validation normalized CRPS \(nCRPS\) across forecasting horizons\.To examine how context\-only forecasting behaves as uncertainty increases, we directly forecast the weather 12 hours and 1, 3, and 5 days ahead without autoregressive rollout\. As shown in[Figure2](https://arxiv.org/html/2609.38632#S3.F2), the validation loss of the CRPS\-trained models generally improves at 12 hours and 1 day\. In contrast, at 3 and 5 days, it initially improves and then deteriorates while the training loss keeps decreasing, indicating overfitting\. Similar patterns appear across diverse architectures, as shown in[Figure10](https://arxiv.org/html/2609.38632#A4.F10)\. These observations suggest that context\-only CRPS training can struggle at longer forecast horizons, prompting us to seek more effective ways to learn the same predictive distributionp⁡\(X∣c\)\{p\(X\\mid c\)\}\.

##### Intuition of Auxiliary Denoising Tasks

Figure 3:Schematic comparison of the context\-only forecasting task and auxiliary conditional denoising tasks\.Figure 4:Effect of conditional denoising tasks on one\-step forecasting\. For 3\-day\-ahead weather forecasting, corrupted futures with noise levelst∈\[tmin,1\]t\\in\[t\_\{\\min\},1\]serve as additional training inputs\. Panels show validation nCRPS \(left\), spread–skill ratio \(SSR, middle\), and test nCRPS change \(%\) relative to the CRPS\-trained baseline \(right\)\. Endpoint\-only training \(tmin=1t\_\{\\min\}=1\) closely matches the CRPS baseline\. Loweringtmint\_\{\\min\}broadens the range of auxiliary tasks, mitigating performance degradation\.Motivated by evidence that joint training on related tasks can improve predictive performance on a target task\([Caruana, 1997](https://arxiv.org/html/2609.38632#bib.bib22)\), we hypothesize that jointly learning to predict the same futureXXwith access to partial information can improve context\-only forecasting through shared parameters\. As illustrated in[Figure3](https://arxiv.org/html/2609.38632#S3.F3), partial future information provides clues about the future to be predicted and can reduce the ambiguity of the auxiliary prediction\. Learning the same context\-future relationship under these conditions may provide useful supervision for the shared model, helping address the difficulties observed in context\-only forecasting\. Shared training may also act as regularization and improve generalization in the original context\-only forecasting task\.

Under this intuition, we seek a family of auxiliary prediction tasks that includes the original forecasting problemp⁡\(X∣c\)p\(X\\mid c\)while allowing control over the amount of additional future information supplied to each task\. The diffusion process in \([4](https://arxiv.org/html/2609.38632#S2.E4)\) naturally satisfies these requirements\. Varying the noise level controls the information ofXtX\_\{t\}, and the independent\-noise endpoint recovers context\-only forecasting\. This yields a family of conditional denoising tasks with target distributions\{p⁡\(X∣c,Xt,t\)\}t∈\[0,1\]\\\{p\(X\\mid c,X\_\{t\},t\)\\\}\_\{t\\in\[0,1\]\}\.

##### Effect of Auxiliary Denoising Tasks

To assess the hypothesis, we jointly train a shared stochastic predictor on such conditional denoising tasks with the objective formalized in[Section3\.2](https://arxiv.org/html/2609.38632#S3.SS2), building on DDMs\([De Bortoli et al\., 2025](https://arxiv.org/html/2609.38632#bib.bib28)\)\. We revisit the 3\-day\-ahead weather forecasting setting and vary the training noise ranget∈\[tmin,1\]t\\in\[t\_\{\\min\},1\], while inference always uses a single forward pass att=1t=1\(NFE=1\\mathrm\{NFE\}=1\)\. As shown in[Figure4](https://arxiv.org/html/2609.38632#S3.F4), endpoint\-only training \(tmin=1t\_\{\\min\}=1\) closely mirrors the validation trajectories of the CRPS\-trained, context\-only baseline\. Loweringtmint\_\{\\min\}provides the model with richer auxiliary supervision, improving one\-step forecasting and mitigating overfitting in line with our hypothesis\. Additional experimental details and results are provided in[SectionD\.2](https://arxiv.org/html/2609.38632#A4.SS2)\.

### 3\.2Proper Scoring Rule\-based Diffusion for Auxiliary Denoising Tasks

We now formalize the joint learning of the conditional predictive distributions introduced in[Section3\.1](https://arxiv.org/html/2609.38632#S3.SS1)\. Building on DDMs\([De Bortoli et al\., 2025](https://arxiv.org/html/2609.38632#bib.bib28)\), we adopt a proper scoring rule\-based objective for a shared stochastic predictor and present its empirical estimator based on finite ensembles\.

##### Proper Scoring Rule\-based Diffusion

To learn the family of the conditional predictive distributions\{p⁡\(X∣c,Xt,t\)\}t∈\[0,1\]\{\\\{p\(X\\mid c,X\_\{t\},t\)\\\}\_\{t\\in\[0,1\]\}\}indexed by the noise leveltt, we minimize the expected scoring loss across noise levels using a shared stochastic predictor\. Specifically, with the auxiliary variableXt=\(1−t\)​X\+t​ε\{X\_\{t\}=\(1\-t\)X\+t\\varepsilon\}andε∼𝒩⁡\(0,I\)\\varepsilon\\sim\\mathcal\{N\}\(0,I\)independent of\(c,X\)\(c,X\)in \([4](https://arxiv.org/html/2609.38632#S2.E4)\), the training objective is

ℒDDMλ,β\(θ\)=∫01wt𝔼c,X,Xt∼qt\(⋅∣X\)\[Sλ,β\(pθ\(⋅∣c,Xt,t\),X\)\]dt,\\mathcal\{L\}\_\{\\mathrm\{DDM\}\}^\{\\lambda,\\beta\}\(\\theta\)=\\int\_\{0\}^\{1\}w\_\{t\}\\,\\mathbb\{E\}\_\{c,X,X\_\{t\}\\sim q\_\{t\}\(\\cdot\\mid X\)\}\\left\[S\_\{\\lambda,\\beta\}\\left\(p\_\{\\theta\}\(\\cdot\\mid c,X\_\{t\},t\),X\\right\)\\right\]\\,\\mathrm\{d\}t,\(6\)wherepθ\(⋅∣c,Xt,t\)p\_\{\\theta\}\(\\cdot\\mid c,X\_\{t\},t\)is the trainable predictive distribution\. The nonnegative weightwtw\_\{t\}controls the contribution of each noise level, and we use uniform weighting \(wt≡1w\_\{t\}\\equiv 1\) throughout this work\.

Here, the scoring ruleSλ,βS\_\{\\lambda,\\beta\}is ageneralized energy score\([De Bortoli et al\., 2025](https://arxiv.org/html/2609.38632#bib.bib28)\)that provides a sample\-based criterion for learning the conditional predictive distribution from observed targets,

Sλ,β​\(P,X\)=𝔼X^∼P​\[‖X^−X‖β\]−λ2​𝔼\(X^,X^′\)∼P⊗P​\[‖X^−X^′‖β\],S\_\{\\lambda,\\beta\}\(P,X\)=\\mathbb\{E\}\_\{\\hat\{X\}\\sim P\}\\left\[\\\|\\hat\{X\}\-X\\\|^\{\\beta\}\\right\]\-\\frac\{\\lambda\}\{2\}\\mathbb\{E\}\_\{\(\\hat\{X\},\\hat\{X\}^\{\\prime\}\)\\sim P\\otimes P\}\\left\[\\\|\\hat\{X\}\-\\hat\{X\}^\{\\prime\}\\\|^\{\\beta\}\\right\],\(7\)whereλ∈\[0,1\]\\lambda\\in\[0,1\]andβ∈\(0,2\]\\beta\\in\(0,2\]control dispersion and distance sensitivity, respectively\. With a proper scoring rule \(λ=1\\lambda=1\), a predictor satisfyingpθ\(⋅∣c,Xt,t\)=p\(⋅∣c,Xt,t\)p\_\{\\theta\}\(\\cdot\\mid c,X\_\{t\},t\)=p\(\\cdot\\mid c,X\_\{t\},t\)for allttattains a population minimum of \([6](https://arxiv.org/html/2609.38632#S3.E6)\)\. At this solution,pθ\(⋅∣c,X1,1\)=p\(⋅∣c\)p\_\{\\theta\}\(\\cdot\\mid c,X\_\{1\},1\)=p\(\\cdot\\mid c\), so each ensemble member can be sampled from the target predictive distribution in a single forward pass, and multi\-step sampling also recovers the same target distribution\. In practice, learning an accurate one\-step predictor can be challenging under high uncertainty, and multi\-step sampling may improve forecasting in such cases\.

##### Empirical Computation

We estimate this objective usingNNtraining pairs\(cn,Xn\)\(c\_\{n\},X\_\{n\}\)andM≥2M\\geq 2predicted samples\{X^n\(m\)\}m=1M\\\{\\hat\{X\}\_\{n\}^\{\(m\)\}\\\}\_\{m=1\}^\{M\}for each corrupted future stateXtnX\_\{t\_\{n\}\}withtn∼𝒰⁡\(0,1\)t\_\{n\}\\sim\\mathcal\{U\}\(0,1\)as

ℒ^DDMλ,β​\(θ\)=1N​∑n=1N\[1M​∑m=1M‖X^n\(m\)−Xn‖β−λ2​M​\(M−1\)​∑m≠m′‖X^n\(m\)−X^n\(m′\)‖β\],\\widehat\{\\mathcal\{L\}\}\_\{\\mathrm\{DDM\}\}^\{\\lambda,\\beta\}\(\\theta\)=\\frac\{1\}\{N\}\\sum\_\{n=1\}^\{N\}\\left\[\\frac\{1\}\{M\}\\sum\_\{m=1\}^\{M\}\\left\\\|\\hat\{X\}\_\{n\}^\{\(m\)\}\-X\_\{n\}\\right\\\|^\{\\beta\}\-\\frac\{\\lambda\}\{2M\(M\-1\)\}\\sum\_\{m\\neq m^\{\\prime\}\}\\left\\\|\\hat\{X\}\_\{n\}^\{\(m\)\}\-\\hat\{X\}\_\{n\}^\{\(m^\{\\prime\}\)\}\\right\\\|^\{\\beta\}\\right\],\(8\)ζn\(m\)∼i\.i\.d\.pζ→introducing stochasticityX^θ\(cn,Xtn,tn,ζn\(m\)\)=X^n\(m\)∼pθ\(⋅∣cn,Xtn,tn\)\.\\zeta\_\{n\}^\{\(m\)\}\\overset\{\\mathrm\{i\.i\.d\.\}\}\{\\sim\}p\_\{\\zeta\}\\;\\xrightarrow\{\\text\{introducing stochasticity\}\}\\;\\hat\{X\}\_\{\\theta\}\(c\_\{n\},X\_\{t\_\{n\}\},t\_\{n\},\\zeta\_\{n\}^\{\(m\)\}\)=\\hat\{X\}\_\{n\}^\{\(m\)\}\\sim p\_\{\\theta\}\(\\cdot\\mid c\_\{n\},X\_\{t\_\{n\}\},t\_\{n\}\)\.\(9\)Here, an additional stochastic sourceζ\\zeta, distinct from the corruption noiseε\\varepsilon, allows the predictor to represent the predictive distribution by generating multiple clean futures under the same conditioning\.

### 3\.3Revisiting CRPS Training from a Diffusion Perspective

We first establish standard context\-only CRPS training as an endpoint\-only special case of our formulation\. We then present the resulting training algorithm, which extends standard CRPS training by incorporating auxiliary conditional denoising tasks while retaining one\-step ensemble generation\.

##### Generalization of CRPS Training

From the distributional diffusion view in[Section3\.2](https://arxiv.org/html/2609.38632#S3.SS2), standard CRPS training can be understood as an endpoint\-only special case\. If we setλ=β=1\\lambda=\\beta=1and apply the score component\-wise, we obtainS1,1=SCRPSS\_\{1,1\}=S\_\{\\texttt\{CRPS\}\}\. With training restricted tot=1t=1and the endpoint predictive distribution independent of corruption noise, Equation[6](https://arxiv.org/html/2609.38632#S3.E6)reduces to the CRPS objective111See[SectionB\.2](https://arxiv.org/html/2609.38632#A2.SS2)for the mathematical details\.:

ℒDDM1,1​\(θ\)\\displaystyle\\mathcal\{L\}\_\{\\mathrm\{DDM\}\}^\{1,1\}\(\\theta\)=∫01𝔼c,X,Xt∼qt\(⋅∣X\)\[SCRPS\(pθ\(⋅∣c,Xt,t\),X\)\]dt\\displaystyle=\\int\_\{0\}^\{1\}\\mathbb\{E\}\_\{c,X,\\,X\_\{t\}\\sim q\_\{t\}\(\\cdot\\mid X\)\}\\left\[S\_\{\\texttt\{CRPS\}\}\\left\(p\_\{\\theta\}\(\\cdot\\mid c,X\_\{t\},t\),X\\right\)\\right\]\\,\\mathrm\{d\}t\(10\)→restrict training to​t=1𝔼c,X\[SCRPS\(pθ\(⋅∣c\),X\)\]\.\\displaystyle\\xrightarrow\{\\text\{restrict training to \}t=1\}\\mathbb\{E\}\_\{c,X\}\\left\[S\_\{\\texttt\{CRPS\}\}\\left\(p\_\{\\theta\}\(\\cdot\\mid c\),X\\right\)\\right\]\.\\vphantom\{\\int\}

From this view, standard CRPS training can be interpreted as learning only the endpoint of the task family in[Section3\.1](https://arxiv.org/html/2609.38632#S3.SS1), and our formulation as generalizing it with auxiliary denoising tasks att<1t<1\.

##### Training Algorithm

Algorithm 1Training

forc,xindataloader:

t = uniform\(0, 1\)

eps = randn\_like\(x\)

x\_t = \(1\-t\)\*x \+ t\*eps

x\_hat=\[\]

forminrange\(M\):

zeta=sample\(p\_zeta\)

pred=model\(c,zeta,x\_t=x\_t,t=t\)

x\_hat\.append\(pred\)

loss=fair\_crps\(x\_hat,x\)

update\(model,loss\)

In this work, we setλ=β=1\{\\lambda=\\beta=1\}and apply the score component\-wise, yielding a CRPS\-based DDM loss that naturally extends standard CRPS training with auxiliary denoising tasks across noise levels\.

The training procedure is summarized in[Section3\.3](https://arxiv.org/html/2609.38632#S3.SS3.SSS0.Px2), withthe only additional operationsrelative to standard CRPS training highlighted inpurple\. These operations sample a noise leveltt, construct a corrupted futureXtX\_\{t\}, and provide both as additional conditioning inputs\. For each context–future pair\(c,X\)\(c,X\), we generateMMensemble members\{X^\(m\)\}m=1M\\\{\\hat\{X\}^\{\(m\)\}\\\}\_\{m=1\}^\{M\}conditioned on the same corrupted futureXtX\_\{t\}, using independent draws ofζ\(m\)\\zeta^\{\(m\)\}to introduce stochasticity\. At each sampled noise level, the predictor is then updated using a loss with the fair CRPS form\.

The resulting training objective averages this fair CRPS overNNcontext–future pairs and noise levels:

L^​\(θ\)=1N​∑n=1N\[1M​∑m=1M‖X^n\(m\)−Xn‖1−12​M​\(M−1\)​∑m≠m′‖X^n\(m\)−X^n\(m′\)‖1\],\\hat\{L\}\(\\theta\)=\\frac\{1\}\{N\}\\sum\_\{n=1\}^\{N\}\\left\[\\frac\{1\}\{M\}\\sum\_\{m=1\}^\{M\}\\left\\\|\\hat\{X\}\_\{n\}^\{\(m\)\}\-X\_\{n\}\\right\\\|\_\{1\}\-\\frac\{1\}\{2M\(M\-1\)\}\\sum\_\{m\\neq m^\{\\prime\}\}\\left\\\|\\hat\{X\}\_\{n\}^\{\(m\)\}\-\\hat\{X\}\_\{n\}^\{\(m^\{\\prime\}\)\}\\right\\\|\_\{1\}\\right\],\(11\)whereX^n\(m\)=X^θ​\(cn,Xtn,tn,ζn\(m\)\)\\hat\{X\}\_\{n\}^\{\(m\)\}=\\hat\{X\}\_\{\\theta\}\(c\_\{n\},X\_\{t\_\{n\}\},t\_\{n\},\\zeta\_\{n\}^\{\(m\)\}\)and∥⋅∥1\\\|\\cdot\\\|\_\{1\}sums absolute differences across forecast channels and spatial locations\. We apply area and channel weighting as detailed in Appendix[C](https://arxiv.org/html/2609.38632#A3), but omit them here for simplicity\. One\-step and multi\-step sampling procedures are provided in Appendix[B\.1](https://arxiv.org/html/2609.38632#A2.SS1)\.

## 4Experiments

We evaluate our formulation against standard CRPS training in two settings\. In the controlled pilot setting, we vary the forecast horizon to assess the contribution of auxiliary conditional denoising tasks under different levels of uncertainty\. In the high\-dimensional global setting, we test whether the gains transfer across training regimes, including training from scratch and fine\-tuning, and examine how the auxiliary tasks affect generalization under distribution shift and in tropical cyclone cases\. Both settings include multiple architectures, allowing us to assess the consistency of the benefits\.

##### Experimental Setup

We use ERA5 reanalysis with1979\-\-2017,2018\-\-2019, and2020for training, validation, and testing, respectively\([Hersbach et al\., 2020](https://arxiv.org/html/2609.38632#bib.bib21)\)\. The controlled experiments use a2\.8125∘2\.8125^\{\\circ\}\(64×12864\\times 128\) global latitude–longitude grid with 9 representative channels, while the large\-scale experiments use a1\.5∘1\.5^\{\\circ\}\(121×240121\\times 240\) grid with 84 channels\. We refer to models trained with the DDM objective as DDM and compare them against standard CRPS training under matched architectures and training configurations\. All results in the main text useNFE=1\\mathrm\{NFE\}=1unless stated otherwise\. Further details including data preprocessing and additional results are described in Appendices[C](https://arxiv.org/html/2609.38632#A3)and[D](https://arxiv.org/html/2609.38632#A4)\.

### 4\.1Pilot Experiments: Analysis of Auxiliary Conditional Denoising Tasks

In this2\.8125∘2\.8125^\{\\circ\}setting, we consider two CRPS\-based model architectures, FGN\([Alet et al\., 2025](https://arxiv.org/html/2609.38632#bib.bib41)\)and FCN3\([Bonev et al\., 2025](https://arxiv.org/html/2609.38632#bib.bib40)\), with direct forecast horizons of12h,1d,3d, and5d\. Each direct forecast horizon serves as therollout interval\(RI\) during autoregressive rollouts\. Detailed experimental configurations and additional results including other models are provided in Appendices[C\.4](https://arxiv.org/html/2609.38632#A3.SS4)and[D\.3](https://arxiv.org/html/2609.38632#A4.SS3)\.

Table 1:Direct Forecasting Performance
Figure 5:Effect of NFE

Table 2:Autoregressive Rollout Performance
Figure 6:SSR Analysis
##### Direct Forecasting Performance

We first assess the contribution of auxiliary conditional denoising tasks to direct forecasting atNFE=1\\text\{NFE\}=1under different levels of uncertainty\. As shown in[Section4\.1](https://arxiv.org/html/2609.38632#S4.SS1), training with the DDM objective consistently yields better performance than standard CRPS training\.

Specifically, the gains are larger in the3dand5dtasks, where predicting future states directly from context is more challenging, than for shorter forecast horizons\. Under high uncertainty, richer supervision from auxiliary conditional denoising tasks may help the shared model better approximate the target predictive distribution\. The noise\-range ablation in[Figure4](https://arxiv.org/html/2609.38632#S3.F4)also supports this interpretation, showing that broadening the range of auxiliary tasks improvesNFE=1\\text\{NFE\}=1forecasting in the3dtask\.

In contrast, gains are smaller in12hand1d\. As noted earlier, both formulations admit population\-optimal solutions that recover the true predictive distributionp⁡\(X∣c\)p\(X\\mid c\)\. In these easier tasks, CRPS training may already approximate the forecast marginals well, limiting potential gains from auxiliary conditional denoising\. This provides one possible explanation for their similar performance\.

##### Effect of Additional NFEs

The benefits of additional NFEs can also be interpreted from the optimality perspective\. At an ideal population optimum where the joint conditional denoising distributions are recovered exactly at all noise levels, DDM sampling recovers the true predictive distributionp⁡\(X∣c\)p\(X\\mid c\)regardless of NFE \(see[SectionB\.1](https://arxiv.org/html/2609.38632#A2.SS1)\)\. When one\-step predictions already approximate this distribution well, additional NFEs may offer limited gains\. Consistent with this expectation,[Section4\.1](https://arxiv.org/html/2609.38632#S4.SS1)shows that NFE gains in the12htask are present only in early training\. In contrast, gains persist in the3dtask, where learning an accurate one\-step predictor may be more difficult and iterative refinement may improve the approximation of the target predictive distribution\.

##### Autoregressive Rollout Performance

We next evaluate autoregressive rollouts atNFE=1\\text\{NFE\}=1per rollout step\. Across both architectures, DDM generally yields modest gains with a12hrollout interval \(RI\) and maintains a clear performance advantage with a3dinterval \([Section4\.1](https://arxiv.org/html/2609.38632#S4.SS1)\)\. At longer RIs, DDM also alleviates the underdispersion observed with CRPS training, maintaining SSRs closer to11and suggesting better ensemble calibration \([Section4\.1](https://arxiv.org/html/2609.38632#S4.SS1)\)\. Together, the pilot results support our expectation that auxiliary conditional denoising tasks improve direct forecasting atNFE=1\\text\{NFE\}=1, with gains that persist through autoregressive rollouts and are more pronounced at longer RIs\.

### 4\.2Scaling to High\-Dimensional Global Weather Forecasting

Towards realistic global weather forecasting, we scale our evaluation to1\.5∘1\.5^\{\\circ\}ERA5 reanalysis with 84 forecast channels\. We consider two recently proposed forecasting models, MOSAIC\([Zhdanov et al\., 2026](https://arxiv.org/html/2609.38632#bib.bib26)\)and U\-Cast\([Cachay et al\., 2026](https://arxiv.org/html/2609.38632#bib.bib27)\), adapting their original training recipes\. The two models provide complementary settings\. MOSAIC is trained probabilistically from scratch and predicts the atmospheric state directly, whereas U\-Cast is probabilistically fine\-tuned from a model pretrained with mean absolute error \(MAE\) and predicts residuals\. They further differ in how stochasticity is introduced, allowing us to evaluate DDM across distinct training regimes and parameterizations\. Detailed descriptions of the model designs and training configurations are provided in Appendix[C\.5](https://arxiv.org/html/2609.38632#A3.SS5)\.

Table 3:CRPS and ensemble\-mean RMSE across representative variables and forecast lead times333Results marked with†are provided for reference, as their configurations differ from our controlled experimental setup\. Our primary comparisons evaluate standard CRPS and DDM training under matched settings\.\.CRPS↓\\downarrowRMSE↓\\downarrow1d3d10d1d3d10dMethodZ500T850Q700Z500T850Q700Z500T850Q700Z500T850Q700Z500T850Q700Z500T850Q700IFS\-ENS†\\dagger24\.10\.3700\.28359\.30\.5500\.427263\.81\.3430\.72545\.80\.7180\.611133\.41\.1050\.902623\.12\.8191\.440GenCast†\\dagger20\.20\.2690\.22254\.30\.4590\.367253\.61\.2810\.67639\.30\.5420\.498123\.50\.9550\.807606\.52\.7371\.375NeuralGCM†\\dagger22\.90\.3330\.26354\.90\.4970\.380253\.41\.2930\.67944\.00\.6580\.543126\.21\.0270\.812606\.82\.7561\.374MOSAIC22\.70\.3000\.23958\.30\.4800\.372261\.01\.3030\.68044\.20\.6010\.525133\.21\.0010\.814619\.92\.7811\.383MOSAICDDM\\textbf\{\{MOSAIC\}\}\_\{\\texttt\{DDM\}\}21\.90\.2900\.23256\.70\.4670\.365257\.61\.2900\.67542\.40\.5810\.513128\.60\.9730\.800610\.12\.7551\.374U\-Cast20\.40\.2810\.22956\.10\.4800\.373260\.11\.3260\.70439\.80\.5630\.507126\.90\.9850\.817615\.72\.7921\.427U\-CastDDM\\textbf\{\{U\-Cast\}\}\_\{\\texttt\{DDM\}\}20\.40\.2800\.22956\.10\.4750\.372259\.51\.3140\.69439\.70\.5620\.507126\.60\.9790\.814613\.22\.7791\.409

##### Forecast Performance

[Footnote3](https://arxiv.org/html/2609.38632#footnote3)compares CRPS and ensemble\-mean RMSE on three variables at 1\-, 3\-, and 10\-day lead times\. AtNFE=1\\text\{NFE\}=1per rollout step, DDM improves both metrics in all cases for MOSAIC and improves or matches them for U\-Cast\.

Figure 7:Relative CRPS difference between DDM\- and CRPS\-trained MOSAIC across all 84 forecast channels over 10\-day rollouts\. Blue indicates better performance of DDM\.These improvements extend beyond the representative variables\. Across all 84 forecast channels, DDM achieves lower CRPS in 787 of 840 channel–lead pairs \(93\.7%\) for MOSAIC, as shown in[Figure7](https://arxiv.org/html/2609.38632#S4.F7)\. For U\-Cast, where the objectives differ only during brief fine\-tuning, the gains are modest but extend across most channel–lead pairs \(82\.0%, see[Figure13](https://arxiv.org/html/2609.38632#A4.F13)\)\. Together, these results indicate that auxiliary conditional denoising tasks benefit high\-dimensional forecasting in both training from scratch and fine\-tuning\.

Figure 8:Ensemble calibration analysis\. \(Top left\) SSR versus forecast lead time for Z500, T850, and Q700\. \(Bottom left\) SSR distributions across all 84 forecast channels at 1\-, 3\-, and 10\-day lead times\. \(Right\) Binned spread–skill plots for the same variables at 1\-, 3\-, and 10\-day lead times\.
##### Calibration Analysis

We next examine whether the forecast\-skill gains from DDM are accompanied by improved ensemble calibration\. The bottom\-left panels of[Figure8](https://arxiv.org/html/2609.38632#S4.F8)show SSR distributions across all 84 forecast channels at 1\-, 3\-, and 10\-day lead times\. DDM shifts these distributions toward11across both architectures and the evaluated lead times, suggesting improved ensemble calibration\.

Binned spread–skill plots assess whether forecasts with larger ensemble spread have larger errors\([Wang and Bishop, 2003](https://arxiv.org/html/2609.38632#bib.bib19);[Delle Monache et al\., 2013](https://arxiv.org/html/2609.38632#bib.bib20)\)\. Forecasts are binned by ensemble spread\. Within each bin, we compare finite\-ensemble\-corrected RMS spread with the ensemble\-mean RMSE\. Proximity to the1:11\{:\}1line indicates agreement between ensemble spread and forecast error across levels of predicted uncertainty, and DDM generally brings the curves closer to this line \([Figure8](https://arxiv.org/html/2609.38632#S4.F8)\)\.

### 4\.3Generalization under Distribution Shift and Extreme Weather

In[Section3\.1](https://arxiv.org/html/2609.38632#S3.SS1), we show that introducing auxiliary conditional denoising mitigates the deterioration in validation performance late in training \([Figure4](https://arxiv.org/html/2609.38632#S3.F4)\)\. Motivated by this observation and the potential regularizing effect of shared learning briefly discussed in[Section3\.1](https://arxiv.org/html/2609.38632#S3.SS1), we assess the generalization ability of DDM\-trained models under distribution shift and in tropical cyclone case studies\.

##### Robustness to Distribution Shift

Table 4:Changes in performance under the ERA5\-to\-HRES distribution shift\.We first evaluate ERA5\-trained MOSAIC models on HRES\-fc0 without fine\-tuning, using the same 732 initialization times in2020\. Across all the channels and 1–10\-day lead times, both models deteriorate relative to ERA5 evaluation, but DDM shows smaller average increases in nCRPS, nRMSE, and mean SSR deviation, as shown in[Table4](https://arxiv.org/html/2609.38632#S4.T4)\. To assess the impact of this degradation, we further compare both models with IFS\-ENS across 19 shared channels at the same lead times\. On ERA5, CRPS\- and DDM\-trained MOSAIC achieve lower CRPS than IFS\-ENS in 99\.5% and 100% of these 190 channel–lead pairs, respectively\. On HRES\-fc0, this fraction drops to 58\.4% for CRPS\-trained MOSAIC, while DDM\-trained MOSAIC still achieves lower CRPS than IFS\-ENS in 86\.8% of pairs\. Full details are provided in[SectionD\.4\.2](https://arxiv.org/html/2609.38632#A4.SS4.SSS2)\.

##### Tropical Cyclone Tracking

![Refer to caption](https://arxiv.org/html/2609.38632v1/mosaic_ifs_era5_hres_with_ian.png)Figure 9:\(Left\) Relative CRPS differences from IFS\-ENS for ERA5\-trained MOSAIC evaluated on ERA5 and HRES\-fc0\. \(Right\) Ensemble track forecasts and IBTrACS best track for Hurricane Ian\.We also evaluate 50\-member ensemble forecasts for Hurricane Ian \(2022\) as an extreme\-weather case study\. We track cyclone centers following[Zhdanov et al\. \(2026\)](https://arxiv.org/html/2609.38632#bib.bib26)and compare the predicted tracks with the IBTrACS best track\([Knapp et al\., 2010](https://arxiv.org/html/2609.38632#bib.bib18)\)\.[Figure9](https://arxiv.org/html/2609.38632#S4.F9)shows predicted tracks and uncertainty cones enclosing 66% of the ensemble track distribution\. In this case, DDM produces track distributions closer to the IBTrACS best track for both architectures\. Additional results including Hurricane Michael and quantitative tracking metrics are provided in[SectionD\.4\.3](https://arxiv.org/html/2609.38632#A4.SS4.SSS3)\.

## 5Conclusion

We introduced a proper scoring rule\-based diffusion framework that incorporates auxiliary conditional denoising tasks, with standard CRPS training recovered as an endpoint\-only special case\. Through controlled experiments, we found that jointly learning the auxiliary tasks improves one\-step forecasting performance and mitigates overfitting, particularly under high uncertainty\. These benefits also extend to high\-dimensional global weather forecasting across different architectures and training regimes, while evaluations under distribution shift and in tropical cyclone cases suggest potential generalization benefits\. Our framework achieves this with only minor conditioning changes to existing CRPS\-based forecasters while retaining one\-step ensemble generation\. However, our work does not establish theoretical guarantees for these benefits, leaving an interesting direction for future work\.

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## Appendix ARelated Work

##### Scoring Rules for Probabilistic Weather Forecasting

Proper scoring rules serve as both evaluation metrics and training objectives for probabilistic forecasting\([Gneiting and Raftery, 2007](https://arxiv.org/html/2609.38632#bib.bib12)\)\. For global weather forecasting, AIFS\-CRPS, FGN, and FourCastNet3 \(FCN3\) train stochastic predictors with CRPS\-based objectives and generate samples without iterative denoising\([Lang et al\., 2026](https://arxiv.org/html/2609.38632#bib.bib11);[Alet et al\., 2025](https://arxiv.org/html/2609.38632#bib.bib41);[Bonev et al\., 2025](https://arxiv.org/html/2609.38632#bib.bib40)\)\. MOSAIC addresses spectral fidelity through its architecture and output parameterization\([Zhdanov et al\., 2026](https://arxiv.org/html/2609.38632#bib.bib26)\), while U\-Cast combines deterministic pretraining with CRPS\-based fine\-tuning and Monte Carlo dropout\([Cachay et al\., 2026](https://arxiv.org/html/2609.38632#bib.bib27)\)\. WeatherNext 3 extends FGN to incorporate observations and predict multiple data modalities\([Rasp et al\., 2026](https://arxiv.org/html/2609.38632#bib.bib43)\)\.

##### Diffusion\-based Weather Forecasting

GenCast and ArchesWeatherGen generate ensemble members with sharp spatial features through iterative diffusion and flow\-based sampling, respectively\([Price et al\., 2025](https://arxiv.org/html/2609.38632#bib.bib8);[Couairon et al\., 2026](https://arxiv.org/html/2609.38632#bib.bib6)\)\. Repeated model evaluations at each forecast step make generating large ensembles over long rollouts computationally expensive\. Efficiency can therefore be improved by parallelizing forecast lead times or reducing the sampling cost of each forecast step\. For the former,[Andrae et al\. \(2025\)](https://arxiv.org/html/2609.38632#bib.bib37)combine lead\-time conditioning with correlated noise to generate temporally coherent ensemble trajectories in parallel\. For the latter, Swift\([Stock et al\., 2026](https://arxiv.org/html/2609.38632#bib.bib24)\)and Tyche\([Xu et al\., 2026](https://arxiv.org/html/2609.38632#bib.bib44)\)enable one\-step generation through consistency and MeanFlow\-based training, respectively, followed by autoregressive CRPS fine\-tuning\. These methods learn a deterministic one\-step transport from noise and introduce CRPS in a separate fine\-tuning stage\. In contrast, our formulation applies the scoring rule at every noise level throughout training and requires only additional conditioning inputs to existing CRPS\-based forecasters\.

##### Auxiliary Denoising for One\-step Forecasting

Distributional diffusion models learn conditional distributions of clean data given corrupted observations across noise levels\([De Bortoli et al\., 2025](https://arxiv.org/html/2609.38632#bib.bib28)\)\. In our forecasting formulation, these prediction tasks retain the same future\-state target while varying the information supplied by its corrupted version\. Multi\-step sampling uses intermediate denoising predictions explicitly, whereas our primary one\-step setting evaluates the predictor only at the fully corrupted endpoint\. In the latter setting, intermediate denoising tasks therefore provide auxiliary supervision through shared parameters rather than additional inference steps\. This interpretation connects to denoising autoencoders, which demonstrate that reconstruction from corrupted inputs can support useful representation learning\([Vincent et al\., 2008](https://arxiv.org/html/2609.38632#bib.bib33);[Vincent et al\., 2010](https://arxiv.org/html/2609.38632#bib.bib2)\), and to multitask learning, which exploits shared representations across related tasks\([Caruana, 1997](https://arxiv.org/html/2609.38632#bib.bib22);[Goodfellow et al\., 2013](https://arxiv.org/html/2609.38632#bib.bib34);[Maurer et al\., 2016](https://arxiv.org/html/2609.38632#bib.bib3)\)\. Together, these perspectives motivate the hypothesis that learning intermediate denoising conditionals can benefit endpoint forecasting through shared learning, potentially acting as an implicit regularizer even when intermediate predictions are not used at inference\.

## Appendix BMathematical Details

### B\.1Sampling Procedures

In this work, we primarily use one\-step sampling \(NFE=1\\text\{NFE\}=1\)\. To explore the potential benefits of additional sampling steps, we follow the procedure described in[De Bortoli et al\. \(2025\)](https://arxiv.org/html/2609.38632#bib.bib28)\.

##### One\-step Sampling

Given forecast contextcc, we generateMMensemble members using the stochastic predictor in \([9](https://arxiv.org/html/2609.38632#S3.E9)\)\. For each member, we independently draw corruption noiseε\(m\)\\varepsilon^\{\(m\)\}from the endpoint distribution of the forward process and stochastic sourceζ0\(m\)∼pζ\\zeta\_\{0\}^\{\(m\)\}\\sim p\_\{\\zeta\}\. The forecast is obtained through

X^\(m\)=X^θ\(c,ε\(m\),1,ζ0\(m\)\),m=1,…,M\.\\hat\{X\}^\{\(m\)\}=\\hat\{X\}\_\{\\theta\}\\left\(c,\\varepsilon^\{\(m\)\},1,\\zeta\_\{0\}^\{\(m\)\}\\right\),\\qquad m=1,\\ldots,M\.\(12\)All members share the same contextccbut use independent draws ofε\\varepsilonandζ\\zeta\.

##### Multi\-step Sampling

ForKK\-step sampling, we use the uniform gridtk=1−k/Kt\_\{k\}=1\-k/Kand initializeXt0\(m\)=ε\(m\)X\_\{t\_\{0\}\}^\{\(m\)\}=\\varepsilon^\{\(m\)\}\. At each step, we independently drawζk\(m\)∼pζ\\zeta\_\{k\}^\{\(m\)\}\\sim p\_\{\\zeta\}and predict a clean future

X^k\(m\)=X^θ​\(c,Xtk\(m\),tk,ζk\(m\)\)\.\\hat\{X\}\_\{k\}^\{\(m\)\}=\\hat\{X\}\_\{\\theta\}\\left\(c,X\_\{t\_\{k\}\}^\{\(m\)\},t\_\{k\},\\zeta\_\{k\}^\{\(m\)\}\\right\)\.\(13\)For the linear noise schedule in[Section3](https://arxiv.org/html/2609.38632#S3), the coupled update is

Xtk\+1\(m\)=X^k\(m\)\+tk\+1tk\(Xtk\(m\)−X^k\(m\)\),k=0,…,K−1\.X\_\{t\_\{k\+1\}\}^\{\(m\)\}=\\hat\{X\}\_\{k\}^\{\(m\)\}\+\\frac\{t\_\{k\+1\}\}\{t\_\{k\}\}\\left\(X\_\{t\_\{k\}\}^\{\(m\)\}\-\\hat\{X\}\_\{k\}^\{\(m\)\}\\right\),\\qquad k=0,\\ldots,K\-1\.\(14\)The contextccremains fixed throughout sampling and the stochastic source is resampled independently across members and steps\. No additional corruption noise is injected between steps, but sampling remains stochastic throughζk\(m\)\\zeta\_\{k\}^\{\(m\)\}\. The final forecast equals the last clean prediction,XtK\(m\)=X^K−1\(m\)X\_\{t\_\{K\}\}^\{\(m\)\}=\\hat\{X\}\_\{K\-1\}^\{\(m\)\}\.

For the heat kernel schedule in[SectionD\.3\.2](https://arxiv.org/html/2609.38632#A4.SS3.SSS2), the update in \([14](https://arxiv.org/html/2609.38632#A2.E14)\) is applied to each spherical\-harmonic coefficient with its degree dependent noise leveltℓ​\(⋅\)t\_\{\\ell\}\(\\cdot\)from \([30](https://arxiv.org/html/2609.38632#A4.E30)\)\. Omitting the member index,

\(Xtk\+1\)ℓ​m=\(X^k\)ℓ​m\+tℓ​\(tk\+1\)tℓ​\(tk\)​\(\(Xtk\)ℓ​m−\(X^k\)ℓ​m\)\.\(X\_\{t\_\{k\+1\}\}\)\_\{\\ell m\}=\(\\hat\{X\}\_\{k\}\)\_\{\\ell m\}\+\\frac\{t\_\{\\ell\}\(t\_\{k\+1\}\)\}\{t\_\{\\ell\}\(t\_\{k\}\)\}\\big\(\(X\_\{t\_\{k\}\}\)\_\{\\ell m\}\-\(\\hat\{X\}\_\{k\}\)\_\{\\ell m\}\\big\)\.\(15\)

##### Exactness under Conditional Distribution Recovery

Letqt\(⋅∣c\)q\_\{t\}\(\\cdot\\mid c\)denote the conditional law ofXt=\(1−t\)​X\+t​εX\_\{t\}=\(1\-t\)X\+t\\varepsilon, withε\\varepsilonindependent of\(c,X\)\(c,X\)\. For any finite grid1=t0\>t1\>⋯\>tK=01=t\_\{0\}\>t\_\{1\}\>\\cdots\>t\_\{K\}=0, assume that the stochastic predictor recovers the true joint conditional distributions,

pθ\(⋅∣c,x,tk\)=p\(⋅∣c,x,tk\),k=0,…,K−1,p\_\{\\theta\}\(\\cdot\\mid c,x,t\_\{k\}\)=p\(\\cdot\\mid c,x,t\_\{k\}\),\\qquad k=0,\\ldots,K\-1,\(16\)where equality holds for almost every contextccandqtk\(⋅∣c\)q\_\{t\_\{k\}\}\(\\cdot\\mid c\)\-almost everyxx\.

For consecutive grid pointst=tk\>s=tk\+1t=t\_\{k\}\>s=t\_\{k\+1\}, letYt∼qt\(⋅∣c\)Y\_\{t\}\\sim q\_\{t\}\(\\cdot\\mid c\)denote the current sampler state andX^∼pθ\(⋅∣c,Yt,t\)\\hat\{X\}\\sim p\_\{\\theta\}\(\\cdot\\mid c,Y\_\{t\},t\)its clean prediction\. The update isYs=\(1−s/t\)​X^\+\(s/t\)​YtY\_\{s\}=\(1\-s/t\)\\hat\{X\}\+\(s/t\)Y\_\{t\}as written above\. Exact conditional sampling restores the joint law of\(X,Xt\)\(X,X\_\{t\}\), and applying this update gives

Law⁡\(X^,Yt∣c\)\\displaystyle\\operatorname\{Law\}\(\\hat\{X\},Y\_\{t\}\\mid c\)=Law⁡\(X,Xt∣c\),\\displaystyle=\\operatorname\{Law\}\(X,X\_\{t\}\\mid c\),\(17\)Law⁡\(Ys∣c\)\\displaystyle\\operatorname\{Law\}\(Y\_\{s\}\\mid c\)=Law⁡\(\(1−s/t\)​X\+\(s/t\)​Xt∣c\)\\displaystyle=\\operatorname\{Law\}\\left\(\(1\-s/t\)X\+\(s/t\)X\_\{t\}\\mid c\\right\)=Law\(\(1−s\)X\+sε∣c\)=qs\(⋅∣c\)\.\\displaystyle=\\operatorname\{Law\}\\left\(\(1\-s\)X\+s\\varepsilon\\mid c\\right\)=q\_\{s\}\(\\cdot\\mid c\)\.Starting fromq1\(⋅∣c\)q\_\{1\}\(\\cdot\\mid c\), induction therefore yields the correct state distribution at every grid point and the final distributionq0\(⋅∣c\)=p\(⋅∣c\)q\_\{0\}\(\\cdot\\mid c\)=p\(\\cdot\\mid c\)\. This holds for any finiteK≥1K\\geq 1, including one\-step sampling, and any corruption\-noise prior independent of\(c,X\)\(c,X\)\. The same argument applies coefficient\-wise to the HK schedule, withs/ts/treplaced bytℓ​\(s\)/tℓ​\(t\)t\_\{\\ell\}\(s\)/t\_\{\\ell\}\(t\)\. Therefore, under exact joint conditional recovery, DDM sampling recovers the target predictive distribution regardless of NFE\.

### B\.2CRPS at the Distributional Diffusion Endpoint

We consider component\-wise CRPS withλ=β=1\\lambda=\\beta=1and assume finite first moments and finite expected scores\. At the endpoint,X1=εX\_\{1\}=\\varepsilonis independent of\(c,X\)\(c,X\), sop\(⋅∣c,X1,1\)=p\(⋅∣c\)p\(\\cdot\\mid c,X\_\{1\},1\)=p\(\\cdot\\mid c\)\.

Endpoint\-only training places all probability mass att=1t=1\. If the predictive distribution satisfiespθ\(⋅∣c,ε,1\)=pθ\(⋅∣c\)p\_\{\\theta\}\(\\cdot\\mid c,\\varepsilon,1\)=p\_\{\\theta\}\(\\cdot\\mid c\)for almost every\(c,ε\)\(c,\\varepsilon\), its objective reduces to standard CRPS training,

ℒend​\(θ\)\\displaystyle\\mathcal\{L\}\_\{\\mathrm\{end\}\}\(\\theta\)=𝔼c,X,ε\[SCRPS\(pθ\(⋅∣c,ε,1\),X\)\]\\displaystyle=\\mathbb\{E\}\_\{c,X,\\varepsilon\}\\left\[S\_\{\\texttt\{CRPS\}\}\\left\(p\_\{\\theta\}\(\\cdot\\mid c,\\varepsilon,1\),X\\right\)\\right\]\(18\)=𝔼c,X\[SCRPS\(pθ\(⋅∣c\),X\)\]\.\\displaystyle=\\mathbb\{E\}\_\{c,X\}\\left\[S\_\{\\texttt\{CRPS\}\}\\left\(p\_\{\\theta\}\(\\cdot\\mid c\),X\\right\)\\right\]\.The first equality defines the endpoint\-only objective, and the second uses independence from corruption noise\. This restricts the predictive distribution and allows stochastic predictions throughζ\\zeta\.

If the predictive distribution depends on the corruption noiseε\\varepsilon, the objectives need not coincide\. To show that they nevertheless share the same population\-optimal forecast marginals when optimized over all predictive distributions, letFθ,j​\(u∣c,ε\)F\_\{\\theta,j\}\(u\\mid c,\\varepsilon\)andFj​\(u∣c\)F\_\{j\}\(u\\mid c\)denote the endpoint predictive and target CDFs evaluated atu∈ℝu\\in\\mathbb\{R\}for each componentj∈\{1,…,d\}j\\in\\\{1,\\ldots,d\\\}, respectively\. The CDF representation of CRPS\([Gneiting and Raftery, 2007](https://arxiv.org/html/2609.38632#bib.bib12)\)yields the population excess risk

ℒend​\(θ\)−ℒ∗=𝔼c,ε​\[∑j=1d∫ℝ\(Fθ,j​\(u∣c,ε\)−Fj​\(u∣c\)\)2​𝑑u\],\\mathcal\{L\}\_\{\\mathrm\{end\}\}\(\\theta\)\-\\mathcal\{L\}^\{\*\}=\\mathbb\{E\}\_\{c,\\varepsilon\}\\left\[\\sum\_\{j=1\}^\{d\}\\int\_\{\\mathbb\{R\}\}\\left\(F\_\{\\theta,j\}\(u\\mid c,\\varepsilon\)\-F\_\{j\}\(u\\mid c\)\\right\)^\{2\}\\,\\mathrm\{d\}u\\right\],\(19\)whereℒ∗=𝔼c,X\[SCRPS\(p\(⋅∣c\),X\)\]\\mathcal\{L\}^\{\*\}=\\mathbb\{E\}\_\{c,X\}\[S\_\{\\texttt\{CRPS\}\}\(p\(\\cdot\\mid c\),X\)\]is independent ofθ\\theta\. The nonnegative excess risk vanishes exactly when all predictive marginals match their targets for almost every\(c,ε\)\(c,\\varepsilon\)\. Therefore, the true conditional forecast marginals minimize both the endpoint\-only DDM and standard CRPS population objectives, even when fixed nonnegative channel and area weights are applied in[SectionC\.1](https://arxiv.org/html/2609.38632#A3.SS1)\.

## Appendix CExperimental Details

### C\.1Data and Preprocessing

##### Data and Splits

We conduct experiments at two spatial resolutions:2\.8125∘2\.8125^\{\\circ\}on a64×12864\\times 128grid for the pilot experiments, and1\.5∘1\.5^\{\\circ\}on a121×240121\\times 240grid for the high\-dimensional experiments\. Both settings use 6\-hourly ERA5 data, with1979\-\-2017for training,2018\-\-2019for validation, and2020for testing\. For checkpoint and configuration selection, we compute validation nCRPS on a fixed set of 240 initializations evenly spaced over the validation period and shared across all runs\.

##### Input and Target Normalization

Letkkindex channels,XXdenote the future state, andXcurX^\{\\mathrm\{cur\}\}denote the current state in the forecast contextcc\. Dynamic input states are standardized channel\-wise using the training\-period meanμk\\mu\_\{k\}and standard deviationσk\\sigma\_\{k\}\. Target normalization depends on the prediction parameterization\. FCN3\([Bonev et al\., 2025](https://arxiv.org/html/2609.38632#bib.bib40)\)and MOSAIC\([Zhdanov et al\., 2026](https://arxiv.org/html/2609.38632#bib.bib26)\)predict the future state, while FGN\([Alet et al\., 2025](https://arxiv.org/html/2609.38632#bib.bib41)\)and U\-Cast\([Cachay et al\., 2026](https://arxiv.org/html/2609.38632#bib.bib27)\)predict its residual relative to the current state\. For each forecast lead, the normalized target is

Tk=\{\(Xk−μk\)/σk,state prediction,\(Xk−Xkcur\)/sk,residual prediction,T\_\{k\}=\\begin\{cases\}\(X\_\{k\}\-\\mu\_\{k\}\)/\\sigma\_\{k\},&\\text\{state prediction\},\\\\\[2\.0pt\] \(X\_\{k\}\-X^\{\\mathrm\{cur\}\}\_\{k\}\)/s\_\{k\},&\\text\{residual prediction\},\\end\{cases\}\(20\)wheresks\_\{k\}is the training\-period standard deviation ofXk−XkcurX\_\{k\}\-X^\{\\mathrm\{cur\}\}\_\{k\}for the corresponding forecast lead\.

##### Spatial and Channel Weighting

All spatially aggregated metrics use cosine\-latitude weights to account for unequal grid\-cell areas\. LetHHandWWdenote the numbers of latitude and longitude grid points, respectively, andϕh\\phi\_\{h\}the latitude of rowhh\. We normalize the area weights to have unit mean:

ah=cos⁡\(ϕh\)H−1​∑h′=1Hcos⁡\(ϕh′\)\.a\_\{h\}=\\frac\{\\cos\(\\phi\_\{h\}\)\}\{H^\{\-1\}\\sum\_\{h^\{\\prime\}=1\}^\{H\}\\cos\(\\phi\_\{h^\{\\prime\}\}\)\}\.\(21\)Then, the spatially and channel\-averaged training loss is

ℒ=1C​H​W​∑k=1C∑h=1H∑w=1Wγk​ah​ℓk​h​w,\\mathcal\{L\}=\\frac\{1\}\{CHW\}\\sum\_\{k=1\}^\{C\}\\sum\_\{h=1\}^\{H\}\\sum\_\{w=1\}^\{W\}\\gamma\_\{k\}\\,a\_\{h\}\\,\\ell\_\{khw\},\(22\)whereℓk​h​w\\ell\_\{khw\}is the grid\-point loss andγk\\gamma\_\{k\}is the channel weight specified below\.

- •In the2\.8125∘2\.8125^\{\\circ\}setting, we use nine representative channels:Z500,T850,Q700,U850,V850,T2M,U10,V10, andMSLP\. All channels are weighted uniformly \(γk=1\\gamma\_\{k\}=1\)\.
- •In the1\.5∘1\.5^\{\\circ\}setting, we use the 84 dynamic state channels listed in Table[5](https://arxiv.org/html/2609.38632#A3.T5), comprising 6 upper\-air variables at 13 pressure levels and 6 surface or near\-surface fields\. Following the pressure\-level weighting convention of[Lam et al\. \(2023\)](https://arxiv.org/html/2609.38632#bib.bib5), each upper\-air channel is assignedγk=p/p¯\\gamma\_\{k\}=p/\\overline\{p\}, wherep¯≈463\.46\\overline\{p\}\\approx 463\.46hPa is the mean of the 13 pressure levels\. For surface and near\-surface fields, we useγk=1\.0\\gamma\_\{k\}=1\.0forT​2​mT2mandγk=0\.1\\gamma\_\{k\}=0\.1for all remaining channels, as summarized in Table[6](https://arxiv.org/html/2609.38632#A3.T6)\.

VariableUnitChannelsSurface and near\-surface fields2\-m temperatureK1Mean sea\-level pressurePa110\-m zonal windm​s−1\\mathrm\{m\\,s^\{\-1\}\}110\-m meridional windm​s−1\\mathrm\{m\\,s^\{\-1\}\}1100\-m zonal windm​s−1\\mathrm\{m\\,s^\{\-1\}\}1100\-m meridional windm​s−1\\mathrm\{m\\,s^\{\-1\}\}1Upper\-air fieldsGeopotentialm2​s−2\\mathrm\{m^\{2\}\\,s^\{\-2\}\}13TemperatureK13Specific humiditykg​kg−1\\mathrm\{kg\\,kg^\{\-1\}\}13Zonal windm​s−1\\mathrm\{m\\,s^\{\-1\}\}13Meridional windm​s−1\\mathrm\{m\\,s^\{\-1\}\}13Vertical velocityPa​s−1\\mathrm\{Pa\\,s^\{\-1\}\}13Total forecast channels84Static and forcing inputs \(not predicted\)Surface geopotentialm2​s−2\\mathrm\{m^\{2\}\\,s^\{\-2\}\}1Land–sea mask–1Latitude \(sin\\sin\)–1Longitude \(sin\\sin,cos\\cos\)–2Local time of day \(sin\\sin,cos\\cos\)–6Time of year \(sin\\sin,cos\\cos\)–6Total static and forcing channels17Table 5:Channels used in the1\.5∘1\.5^\{\\circ\}experiments\.

Table 6:Channel weights in1\.5∘1\.5^\{\\circ\}experiments\.

### C\.2Evaluation Metrics

We evaluateNNinitializations indexed byii\. For initializationii, a forecast lead, and channelkk, let\{X^i\(m\)\}m=1M\\\{\\hat\{X\}\_\{i\}^\{\(m\)\}\\\}\_\{m=1\}^\{M\}denote the ensemble forecasts withM≥2M\\geq 2,X¯i=M−1​∑m=1MX^i\(m\)\\bar\{X\}\_\{i\}=M^\{\-1\}\\sum\_\{m=1\}^\{M\}\\hat\{X\}\_\{i\}^\{\(m\)\}their mean, andXiX\_\{i\}the verifying state\. We omit the lead index throughout and the channel indexkkwhen unambiguous\. Predictions are evaluated in state space after reconstructing states, and we use the normalized cosine\-latitude weightsaha\_\{h\}defined in[SectionC\.1](https://arxiv.org/html/2609.38632#A3.SS1)\.

##### CRPS

We average the fair CRPS estimates over initializations and grid points with area weights,

CRPS=1N∑i=1N1H​W∑h=1H∑w=1Wah\[\\displaystyle\\mathrm\{CRPS\}=\\frac\{1\}\{N\}\\sum\_\{i=1\}^\{N\}\\frac\{1\}\{HW\}\\sum\_\{h=1\}^\{H\}\\sum\_\{w=1\}^\{W\}a\_\{h\}\\Bigg\[1M​∑m=1M\|X^i,h,w\(m\)−Xi,h,w\|\\displaystyle\\frac\{1\}\{M\}\\sum\_\{m=1\}^\{M\}\\left\|\\hat\{X\}\_\{i,h,w\}^\{\(m\)\}\-X\_\{i,h,w\}\\right\|\(23\)−12​M​\(M−1\)∑m≠m′\|X^i,h,w\(m\)−X^i,h,w\(m′\)\|\]\.\\displaystyle\}\{\\displaystyle\-\\frac\{1\}\{2M\(M\-1\)\}\\sum\_\{m\\neq m^\{\\prime\}\}\\left\|\\hat\{X\}\_\{i,h,w\}^\{\(m\)\}\-\\hat\{X\}\_\{i,h,w\}^\{\(m^\{\\prime\}\)\}\\right\|\\Bigg\]\.

##### Ensemble\-mean RMSE

Following WeatherBench 2\([Rasp et al\., 2024](https://arxiv.org/html/2609.38632#bib.bib4)\), we average the area\-weighted squared error of the ensemble mean over initializations before taking the square root,

RMSE=1N​∑i=1N1H​W​∑h=1H∑w=1Wah​\(X¯i,h,w−Xi,h,w\)2\.\\mathrm\{RMSE\}=\\sqrt\{\\frac\{1\}\{N\}\\sum\_\{i=1\}^\{N\}\\frac\{1\}\{HW\}\\sum\_\{h=1\}^\{H\}\\sum\_\{w=1\}^\{W\}a\_\{h\}\\bigl\(\\bar\{X\}\_\{i,h,w\}\-X\_\{i,h,w\}\\bigr\)^\{2\}\}\.\(24\)

##### Spread–skill Ratio \(SSR\)

For each initialization, we compute the unbiased ensemble variance

vi,h,w=1M−1​∑m=1M\(X^i,h,w\(m\)−X¯i,h,w\)2v\_\{i,h,w\}=\\frac\{1\}\{M\-1\}\\sum\_\{m=1\}^\{M\}\\left\(\\hat\{X\}\_\{i,h,w\}^\{\(m\)\}\-\\bar\{X\}\_\{i,h,w\}\\right\)^\{2\}\(25\)and the corresponding spatial RMS spread

Spreadi=1H​W​∑h=1H∑w=1Wah​vi,h,w\.\\mathrm\{Spread\}\_\{i\}=\\sqrt\{\\frac\{1\}\{HW\}\\sum\_\{h=1\}^\{H\}\\sum\_\{w=1\}^\{W\}a\_\{h\}v\_\{i,h,w\}\}\.\(26\)As for RMSE, we pool squared spread over initializations before taking the square root\. Following[Bonev et al\. \(2025\)](https://arxiv.org/html/2609.38632#bib.bib40), the finite\-ensemble\-corrected SSR for channelkkis

SSRk=M\+1M​N−1​∑i=1NSpreadi,k2RMSEk\.\\mathrm\{SSR\}\_\{k\}=\\sqrt\{\\frac\{M\+1\}\{M\}\}\\,\\frac\{\\sqrt\{N^\{\-1\}\\sum\_\{i=1\}^\{N\}\\mathrm\{Spread\}\_\{i,k\}^\{\\,2\}\}\}\{\\mathrm\{RMSE\}\_\{k\}\}\.\(27\)The correction accounts for the sampling variability of the finite ensemble mean\. When reporting an aggregate across channels, we average the channel\-wise SSRs\.

##### Normalized CRPS and Ensemble\-mean RMSE

LetCRPSk\\mathrm\{CRPS\}\_\{k\}andRMSEk\\mathrm\{RMSE\}\_\{k\}denote the metrics above for channelkk\. Their normalized aggregates are

nCRPS=1C​∑k=1CCRPSkσk,nRMSE=1C​∑k=1CRMSEkσk\.\\mathrm\{nCRPS\}=\\frac\{1\}\{C\}\\sum\_\{k=1\}^\{C\}\\frac\{\\mathrm\{CRPS\}\_\{k\}\}\{\\sigma\_\{k\}\},\\qquad\\mathrm\{nRMSE\}=\\frac\{1\}\{C\}\\sum\_\{k=1\}^\{C\}\\frac\{\\mathrm\{RMSE\}\_\{k\}\}\{\\sigma\_\{k\}\}\.\(28\)Hereσk\\sigma\_\{k\}is the training\-period state normalization standard deviation\. These summaries weight channels equally without the training loss weightsγk\\gamma\_\{k\}\. In the case of training curves in[Figure10](https://arxiv.org/html/2609.38632#A4.F10), nCRPS uses the target normalization of each model, which may differ from the state normalization\.

### C\.3Architectural Modifications

Throughout this work, we adapt the forecasting backbones to DDM by adding conditioning on the corrupted targetXtX\_\{t\}and noise leveltt\. These modifications retain the prediction parameterization and ensemble\-generation mechanism of the corresponding CRPS\-based weather forecasting models\. Thus, the architectural changes are limited to expanding the input layers and adding time\-conditioning modules, resulting in the modest parameter increases reported in[Tables7](https://arxiv.org/html/2609.38632#A3.T7)and[9](https://arxiv.org/html/2609.38632#A3.T9)\.

### C\.4Experimental Details at 2\.8125 degrees

##### Architectures

The pilot study spans four backbones\. Two are general\-purpose architectures: Swin Transformer \(Swin\)\([Liu et al\., 2021](https://arxiv.org/html/2609.38632#bib.bib30)\), a hierarchical window\-attention transformer, andU\-Net\([Ronneberger et al\., 2015](https://arxiv.org/html/2609.38632#bib.bib31)\), a convolutional encoder–decoder with skip connections\. The other two are CRPS\-based probabilistic weather forecasting models:FGN\([Alet et al\., 2025](https://arxiv.org/html/2609.38632#bib.bib41)\), a graph\-based model that injects noise through a low\-dimensional latent vector, andFCN3\([Bonev et al\., 2025](https://arxiv.org/html/2609.38632#bib.bib40)\), a spherical neural operator that injects noise internally at multiple spatial scales\. Each backbone is trained under both objectives with matched optimization settings\.

Table 7:Backbones used in the2\.8125∘2\.8125^\{\\circ\}pilot study\.[Table7](https://arxiv.org/html/2609.38632#A3.T7)summarizes the model backbones, their prediction targets and noise\-entry points, and the parameter counts\. Switching from CRPS to DDM only expands the input layers to accept the corrupted targetXtX\_\{t\}and adds time\-conditioning modules fortt\.

##### Training Configuration

The main pilot experiments use the optimization settings in[Table8](https://arxiv.org/html/2609.38632#A3.T8)\. Each training step drawsM=2M=2ensemble members and scores them with the fair CRPS estimator\. Evaluation uses EMA weights\. Each run targets a single forecast lead \(12h,1d,3d,5d\{\\color\[rgb\]\{0\.7695,0\.3047,0\.3203\}\\textbf\{\{12h\}\}\},\{\\color\[rgb\]\{0\.9023,0\.625,0\}\\textbf\{\{1d\}\}\},\{\\color\[rgb\]\{0\.1328,0\.5352,0\.1992\}\\textbf\{\{3d\}\}\},\{\\color\[rgb\]\{0,0\.6523,0\.6523\}\\textbf\{\{5d\}\}\}\), taking its two input frames separated by the lead \(FCN3 uses a single frame, following its native configuration\)\. Widths and depths are set per backbone to a comparable scale and are held fixed across the objectives\.

Table 8:Optimization settings, shared by every2\.8125∘2\.8125^\{\\circ\}pilot run\.

### C\.5Experimental Details at 1\.5 degrees

##### Architectures

At1\.5∘1\.5^\{\\circ\}, we use the official implementations of MOSAIC\([Zhdanov et al\., 2026](https://arxiv.org/html/2609.38632#bib.bib26)\)and U\-Cast\([Cachay et al\., 2026](https://arxiv.org/html/2609.38632#bib.bib27)\)to compare CRPS training with DDM\.[Table9](https://arxiv.org/html/2609.38632#A3.T9)summarizes the prediction targets, ensemble\-generation mechanisms, and parameter counts\. DDM retains each backbone’s original prediction parameterization and ensemble\-generation mechanism, while additionally conditioning on the corrupted prediction target and its noise level\. This conditioning increases the parameter count by3\.23%3\.23\\%for MOSAIC and1\.06%1\.06\\%for U\-Cast\.

Table 9:Backbones used in the1\.5∘1\.5^\{\\circ\}experiments\.
##### Training Configuration

Under our common experimental setup, we follow the published training recipes of MOSAIC and U\-Cast, using only the first training stage for MOSAIC and omitting deep ensembling for U\-Cast\. The training settings summarized in[Table10](https://arxiv.org/html/2609.38632#A3.T10)are matched between the CRPS and DDM variants of each backbone\. Based on the pilot results in[SectionD\.3\.2](https://arxiv.org/html/2609.38632#A4.SS3.SSS2), both DDM variants use spherical Gaussian corruption noise \(SpG\) with the heat\-kernel schedule \(κ0=0\.01\\kappa\_\{0\}=0\.01\)\.

Table 10:Training configurations at1\.5∘1\.5^\{\\circ\}\.

## Appendix DAdditional Results

### D\.1Training Dynamics of Context\-Only CRPS Training

We extend the analysis in[Section3\.1](https://arxiv.org/html/2609.38632#S3.SS1)by examining training and validation trajectories across all four pilot architectures\. As shown in[Figure10](https://arxiv.org/html/2609.38632#A4.F10), validation nCRPS decreases and gradually levels off at the 12\-hour and 1\-day horizons\. At the 3\- and 5\-day horizons, it reaches an early minimum and then increases while training loss continues to decrease\. This increase is larger at 5 days than at 3 days across all four architectures, indicating more pronounced overfitting at longer forecast horizons\.

Training loss uses residual normalization for FGN, Swin, and U\-Net and state normalization for FCN3, consistent with the prediction targets in[Table7](https://arxiv.org/html/2609.38632#A3.T7)\. Validation nCRPS uses state normalization for all models\. The two losses are therefore on different scales for all models except FCN3, and we assess overfitting from their trends during training\. Training configurations are provided in[SectionC\.4](https://arxiv.org/html/2609.38632#A3.SS4)\.

Figure 10:Training loss \(solid, right axis\) and validation nCRPS \(dashed, left axis\) of CRPS\-trained FGN, FCN3, Swin, and U\-Net across forecast horizons at2\.8125∘2\.8125^\{\\circ\}\. Training loss uses residual normalization for FGN, Swin, and U\-Net and state normalization for FCN3, whereas validation nCRPS uses state normalization for all models\. Training and validation losses therefore share the same normalization only for FCN3\. Training configurations are provided in[SectionC\.4](https://arxiv.org/html/2609.38632#A3.SS4)\.
### D\.2Proof\-of\-Concept Experiment in[Section3\.1](https://arxiv.org/html/2609.38632#S3.SS1)

We provide additional results for the proof\-of\-concept experiment in[Section3\.1](https://arxiv.org/html/2609.38632#S3.SS1), including complete test metrics for FCN3 and an extended experiment with FGN\. Both models perform direct 3\-day\-ahead forecasting at2\.8125∘2\.8125^\{\\circ\}resolution\. We compare the CRPS baseline with DDM trained usingt∼𝒰⁡\[tmin,1\]t\\sim\\mathcal\{U\}\[t\_\{\\min\},1\]fortmin∈\{0,0\.25,0\.50,0\.75\}t\_\{\\min\}\\in\\\{0,0\.25,0\.50,0\.75\\\}, together with endpoint\-only training at exactlyt=1t=1\. We train FCN3 for 50 epochs and extend FGN training from 50 to 100 epochs to better observe later training behavior\. For each run, we select the checkpoint with the lowest validation nCRPS, evaluated using 240 initializations from2018\-\-2019withM=50M=50ensemble members\.

[Tables11](https://arxiv.org/html/2609.38632#A4.T11)and[12](https://arxiv.org/html/2609.38632#A4.T12)report test nCRPS and ensemble\-mean nRMSE, with percentage changes relative to the corresponding CRPS baseline\. Endpoint\-only training \(tmin=1t\_\{\\min\}=1\) closely matches CRPS training, reducing nCRPS by0\.47%0\.47\\%for FCN3 and0\.41%0\.41\\%for FGN\. As discussed in[SectionB\.2](https://arxiv.org/html/2609.38632#A2.SS2), endpoint\-only and standard CRPS training share the same population\-optimal forecast marginals\. Although the learned predictive distribution need not be independent of corruption noise, the similar performance observed here is consistent with this theoretical relationship\. Loweringtmint\_\{\\min\}progressively improves both metrics for both architectures\. With the full auxiliary conditional denoising tasks \(tmin=0t\_\{\\min\}=0\), nCRPS and nRMSE decrease by8\.10%8\.10\\%and6\.55%6\.55\\%for FCN3, and by9\.96%9\.96\\%and8\.55%8\.55\\%for FGN\.

The validation trajectories in[Figures4](https://arxiv.org/html/2609.38632#S3.F4)and[11](https://arxiv.org/html/2609.38632#A4.F11)provide a complementary view\. For CRPS and endpoint\-only training, validation nCRPS increases after an initial improvement, while SSR declines substantially below11\. Broader training noise ranges mitigate this degradation and maintain SSR closer to11, although underdispersion remains\. This qualitative pattern is observed for both FCN3 and FGN\. Together, these results are consistent with a regularizing effect of auxiliary conditional denoising tasks learned through shared parameters, supporting the hypothesis that these tasks can improve one\-step forecasting and provide better generalization\.

Figure 11:Effect of conditional denoising tasks on FGN one\-step forecasting\.FGN is trained for 100 epochs for 3\-day\-ahead forecasting at2\.8125∘2\.8125^\{\\circ\}\. Panels show validation nCRPS \(left\), validation SSR \(middle\), and test nCRPS changes \(%\) relative to the CRPS baseline \(right\)\. Test results use checkpoints selected by validation nCRPS, withNFE=1\\mathrm\{NFE\}=1for all DDM configurations\.Table 11:Effect oftmint\_\{\\min\}on FCN3\-3done\-step forecasting performance at2\.8125∘2\.8125^\{\\circ\}used in[Figure4](https://arxiv.org/html/2609.38632#S3.F4)\.Table 12:Effect oftmint\_\{\\min\}on FGN\-3d\-100ep one\-step forecasting at2\.8125∘2\.8125^\{\\circ\}used in[Figure11](https://arxiv.org/html/2609.38632#A4.F11)\. Note that the best checkpoint of the CRPS baseline is the same as FGN\-3d\-50ep due to overfitting\.
### D\.3Additional Results for the2\.8125∘2\.8125^\{\\circ\}Pilot Experiments

#### D\.3\.1Direct Forecasting and Autoregressive Rollout Performance

[Table13](https://arxiv.org/html/2609.38632#A4.T13)reports one\-step forecasting performance for all four backbones, and[Tables14](https://arxiv.org/html/2609.38632#A4.T14)and[15](https://arxiv.org/html/2609.38632#A4.T15)extend the rollout evaluation of FGN and FCN3\. As in[Sections4\.1](https://arxiv.org/html/2609.38632#S4.SS1)and[4\.1](https://arxiv.org/html/2609.38632#S4.SS1), DDM generally improves nCRPS and nRMSE, with larger gains at longer rollout intervals that partly persist during rollouts\.

Table 13:Direct forecasting performance at2\.8125∘2\.8125^\{\\circ\}across four architectures\.Table 14:Autoregressive rollout nCRPS at2\.8125∘2\.8125^\{\\circ\}\.Table 15:Autoregressive rollout nRMSE at2\.8125∘2\.8125^\{\\circ\}\. The same runs and conventions as[Table14](https://arxiv.org/html/2609.38632#A4.T14)\.
#### D\.3\.2Noise Prior and Schedule Ablation

To assess the sensitivity of one\-step forecasting to the corruption process, we compare spatial noise priors and corruption schedules while keeping the remaining DDM settings fixed \([Table16](https://arxiv.org/html/2609.38632#A4.T16)\)\. Here,xxis the normalized prediction target andε\\varepsilonis corruption noise, distinct from the randomness sourceζ\\zeta\.

Standard diffusion models typically use independent Gaussian noise at each grid point \(Grid\)\([Ho et al\., 2020](https://arxiv.org/html/2609.38632#bib.bib29)\)\. GenCast reports small but consistent improvements from sampling isotropic Gaussian noise in the spherical\-harmonic domain\([Price et al\., 2025](https://arxiv.org/html/2609.38632#bib.bib8)\)\. Motivated by this finding, we evaluate spherical Gaussian \(SpG\) noise with equal variance across retained harmonic modes, generated as

εSpG​\(ω\)=g​∑ℓ=0Lnoise∑m=−ℓℓaℓ​m​ψℓ​m​\(ω\),aℓ​m​∼i\.i\.d\.​𝒩​\(0,1\),\\varepsilon\_\{\\mathrm\{SpG\}\}\(\\omega\)=g\\sum\_\{\\ell=0\}^\{L\_\{\\mathrm\{noise\}\}\}\\sum\_\{m=\-\\ell\}^\{\\ell\}a\_\{\\ell m\}\\,\\psi\_\{\\ell m\}\(\\omega\),\\qquad a\_\{\\ell m\}\\overset\{\\mathrm\{i\.i\.d\.\}\}\{\\sim\}\\mathcal\{N\}\(0,1\),\(29\)whereω∈𝕊2\\omega\\in\\mathbb\{S\}^\{2\},ψℓ​m\\psi\_\{\\ell m\}are real orthonormal spherical harmonics,LnoiseL\_\{\\mathrm\{noise\}\}is the maximum degree, andggis a normalization constant\. We sample this isotropic Gaussian field independently across channels\.

Figure 12:Effective noise levels by harmonic degree forκ0=0\.01\\kappa\_\{0\}=0\.01\(top\) andκ0=0\.05\\kappa\_\{0\}=0\.05\(bottom\)\.Linear interpolation attenuates all target coefficients by the same factor\. We therefore consider a spherical heat\-kernel \(HK\) schedule, inspired by[Hoogeboom and Salimans \(2023\)](https://arxiv.org/html/2609.38632#bib.bib32), that attenuates fine\-scale target information more rapidly\.

Since spherical harmonics of degreeℓ\\ellhave Laplace–Beltrami eigenvalue−ℓ⁡\(ℓ\+1\)\-\\ell\(\\ell\+1\)on the unit sphere\([Atkinson and Han, 2012](https://arxiv.org/html/2609.38632#bib.bib1)\), we define the interpolation for degrees0≤ℓ≤LHK0\\leq\\ell\\leq L\_\{\\mathrm\{HK\}\}as

tℓ​\(t\)\\displaystyle t\_\{\\ell\}\(t\)=1−\(1−t\)​exp⁡\[−κ0​t2​ℓ​\(ℓ\+1\)\],\\displaystyle=1\-\(1\-t\)\\exp\\\!\\left\[\-\\frac\{\\kappa\_\{0\}t\}\{2\}\\ell\(\\ell\+1\)\\right\],\(30\)\(xt\)ℓ​m\\displaystyle\(x\_\{t\}\)\_\{\\ell m\}=\[1−tℓ​\(t\)\]​xℓ​m\+tℓ​\(t\)​εℓ​m\.\\displaystyle=\\bigl\[1\-t\_\{\\ell\}\(t\)\\bigr\]x\_\{\\ell m\}\+t\_\{\\ell\}\(t\)\\varepsilon\_\{\\ell m\}\.
Here,xℓ​mx\_\{\\ell m\}andεℓ​m\\varepsilon\_\{\\ell m\}are the spherical\-harmonic coefficients of the target and noise,LHKL\_\{\\mathrm\{HK\}\}is the maximum degree, andκ0≥0\\kappa\_\{0\}\\geq 0controls the attenuation strength\. As shown in[Figure12](https://arxiv.org/html/2609.38632#A4.F12), higher degrees have largertℓ​\(t\)t\_\{\\ell\}\(t\)and smaller target weights1−tℓ​\(t\)1\-t\_\{\\ell\}\(t\)at the samett\. Largerκ0\\kappa\_\{0\}strengthens this scale dependence\.

In practice,ggis computed analytically to ensure unit grid\-averaged marginal variance\. We use\(Lnoise,LHK\)=\(63,31\)\(L\_\{\\mathrm\{noise\}\},L\_\{\\mathrm\{HK\}\}\)=\(63,31\)at2\.8125∘2\.8125^\{\\circ\}and\(120,60\)\(120,60\)at1\.5∘1\.5^\{\\circ\}\. The residual outside the retained representation is preserved and interpolated usingtLHK​\(t\)t\_\{L\_\{\\mathrm\{HK\}\}\}\(t\)\. TheGridandSpGvariants use linear interpolation, with independent standard Gaussian grid noise and spherical Gaussian noise, respectively\. TheHKvariants use the same spherical Gaussian prior withκ0∈\{0\.01,0\.05\}\\kappa\_\{0\}\\in\\\{0\.01,0\.05\\\}\.

[Table16](https://arxiv.org/html/2609.38632#A4.T16)shows thatHKgenerally achieves the lowest nCRPS at12hand1d, whereasGridorSpGperforms best at3dand5d\. Thus, the preferred noise distribution and schedule depend on the rollout interval and backbone\. Nevertheless, all DDM configurations outperform the CRPS baseline at the two longer intervals across all four backbones\. For the main pilot experiments, we select the noise prior and schedule with the lowest validation nCRPS for each backbone and forecast horizon\.

Table 16:Noise scheduling for each backbone and RI at2\.8125∘2\.8125^\{\\circ\}, with the single\-step nCRPS\. Values are test nCRPS\. Shaded cells denote the configuration selected by the lowest validation nCRPS\.

### D\.4Additional Results at1\.5∘1\.5^\{\\circ\}High\-dimensional Global Weather Forecasting

#### D\.4\.1Forecast Performance

We provide channel\-wise comparisons and rollout curves using all 732 initializations at 00 and 12 UTC in2020, withM=50M=50ensemble members andNFE=1\\text\{NFE\}=1for DDM\. Figure[13](https://arxiv.org/html/2609.38632#A4.F13)compares DDM and CRPS training across all 84 forecast channels over 10\-day rollouts\. In addition to IFS\-ENS,[Footnote3](https://arxiv.org/html/2609.38632#footnote3)reports results for GenCast\([Price et al\., 2025](https://arxiv.org/html/2609.38632#bib.bib8)\)and NeuralGCM\([Kochkov et al\., 2024](https://arxiv.org/html/2609.38632#bib.bib7)\)\. These baselines are included for reference, as their configurations, including spatial resolution, differ from ours\. Our primary focus is comparing DDM and CRPS training under matched experimental settings\.

Comparisons use native forecast intervals of 24 h for MOSAIC and 12 h for U\-Cast, yielding 840 and 1,680 channel–lead pairs, respectively\. DDM achieves lower CRPS in93\.7%93\.7\\%\(787/840\) of these pairs for MOSAIC and82\.0%82\.0\\%\(1,377/1,680\) for U\-Cast\. The corresponding fractions for ensemble\-mean RMSE are92\.0%92\.0\\%\(773/840\) and81\.0%81\.0\\%\(1,360/1,680\)\. Improvements extend across surface variables and most tropospheric levels, whereas degradations are concentrated at the lowest\-pressure levels\.

We next visualize autoregressive rollout curves on 19 forecast channels common to our models and the publicly available WeatherBench 2 IFS\-ENS archive\([Rasp et al\., 2024](https://arxiv.org/html/2609.38632#bib.bib4)\)\(Figures[14](https://arxiv.org/html/2609.38632#A4.F14)–[16](https://arxiv.org/html/2609.38632#A4.F16)\)\. For both backbones, DDM achieves lower CRPS than the corresponding CRPS\-trained model at every evaluated channel–lead pair in this comparison, although several U\-Cast differences fall below the precision displayed in[Footnote3](https://arxiv.org/html/2609.38632#footnote3)\. Ensemble\-mean RMSE improves across all pairs for MOSAIC and nearly all pairs for U\-Cast\. The SSR curves show that DDM generally moves the SSR closer to11\.

We additionally evaluate MOSAIC withNFE=2\\mathrm\{NFE\}=2under the same evaluation protocol\. As shown in[Figure17](https://arxiv.org/html/2609.38632#A4.F17), two\-step sampling retains lower CRPS in all 787 channel–lead pairs where one\-step sampling outperforms CRPS\-trained MOSAIC and achieves lower CRPS in 16 additional pairs, bringing the total to 803 out of 840\. These results illustrate the potential benefits of refinement without retraining, at the cost of one additional model evaluation per member and rollout step\.

Figure 13:Relative performance differences in CRPS \(left\) and ensemble\-mean RMSE \(right\) between DDM and CRPS training for MOSAIC \(top\) and U\-Cast \(bottom\)\. Negative values \(blue\) indicate that DDM shows better performance than CRPS\.Figure 14:CRPS over 10\-day autoregressive rollouts in ERA52020\.Figure 15:Ensemble\-mean RMSE over 10\-day autoregressive rollouts in ERA52020\.Figure 16:SSR over 10\-day autoregressive rollouts in ERA52020\.Figure 17:Effect of NFE on MOSAIC forecasting performance\. Relative CRPS differences between DDM and CRPS\-trained MOSAIC across 84 forecast channels over 10\-day autoregressive rollouts\. The same DDM checkpoint is evaluated withNFE=1\\text\{NFE\}=1\(left\) andNFE=2\\text\{NFE\}=2\(right\) in ERA52020\.
#### D\.4\.2Robustness to Distribution Shift

In the distribution shift experiment, we initialize and verify forecasts with HRES\-fc0\. As in the ERA5 evaluation, we use all 732 initializations at 00 and 12 UTC in2020, withM=50M=50ensemble members andNFE=1\\text\{NFE\}=1for DDM\. Note that the SSR deviation reported in[Table4](https://arxiv.org/html/2609.38632#S4.T4)refers to\|SSR−1\|\|\\mathrm\{SSR\}\-1\|averaged over all channels and lead times, with smaller values indicating better spread–skill balance\.

To compare with the IFS\-ENS baseline, Figures[18](https://arxiv.org/html/2609.38632#A4.F18)–[20](https://arxiv.org/html/2609.38632#A4.F20)show rollout curves for CRPS, ensemble\-mean RMSE and SSR for the 19 channels shared with IFS\-ENS\. DDM achieves lower CRPS and ensemble\-mean RMSE than the CRPS\-trained model across all 190 channel–lead pairs and the spread–skill ratio is also generally closer to11, indicating better agreement between ensemble spread and forecast error\. These results show that the advantages of DDM persist under the ERA5–HRES shift\.

Figure 18:CRPS over 10\-day autoregressive rollouts in HRES\-fc02020\.Figure 19:Ensemble\-mean RMSE over 10\-day autoregressive rollouts in HRES\-fc02020\.Figure 20:SSR over 10\-day autoregressive rollouts in HRES\-fc02020\.
#### D\.4\.3Tropical Cyclone Tracking

We provide additional tropical\-cyclone tracking results for Hurricanes Ian and Michael using 50\-member ensembles from MOSAIC and U\-Cast, with IBTrACS best tracks as the reference\. Figure[21](https://arxiv.org/html/2609.38632#A4.F21)shows the ensemble track forecasts for Hurricane Michael, and Table[17](https://arxiv.org/html/2609.38632#A4.T17)reports ensemble\-mean track error and track CRPS for both storms\. DDM improves both metrics over CRPS training\.

To examine the ensemble forecast positions beyond the ensemble mean, Figures[22](https://arxiv.org/html/2609.38632#A4.F22)–[24](https://arxiv.org/html/2609.38632#A4.F24)show enlarged views of Hurricane Ian forecasts with circles of radius 150, 200, and 300 km around the time\-matched IBTrACS center\. The highlighted lead times are 48 h for MOSAIC and 120 h for U\-Cast\. Table[18](https://arxiv.org/html/2609.38632#A4.T18)summarizes the number and percentage of members within each radius, using all 50 members regardless of whether individual positions fall inside the displayed crop\. Across these radii, DDM places more members near the reference center than CRPS training for both backbones\. At 200 km, the fraction increases from62%62\\%to94%94\\%for MOSAIC and from36%36\\%to46%46\\%for U\-Cast\.

##### Tracking and Metrics

We track storm centers using MSLP minima following[Zhdanov et al\. \(2026\)](https://arxiv.org/html/2609.38632#bib.bib26)\. Each member track is initialized at the IBTrACS\([Knapp et al\., 2010](https://arxiv.org/html/2609.38632#bib.bib18)\)position and subsequently follows the minimum MSLP within a search radius around its previous center\. Forecasts are evaluated against the time\-matched IBTrACS best track during the tropical phase\.

At each lead time, let\{𝐱m\}m=1M\\\{\\mathbf\{x\}\_\{m\}\\\}\_\{m=1\}^\{M\}denote the predicted storm positions and𝐲\\mathbf\{y\}the observed position\. We measure great\-circle distanceddusing the haversine formula with Earth radiusR=6371R=6371km\. The ensemble\-mean track error isd⁡\(𝐱¯,𝐲\)d\(\\bar\{\\mathbf\{x\}\},\\mathbf\{y\}\), where𝐱¯\\bar\{\\mathbf\{x\}\}is the arithmetic mean of the member latitude–longitude coordinates\. Following TCBench\([Gomez et al\., 2026](https://arxiv.org/html/2609.38632#bib.bib42)\), we compute track CRPS as

TrackCRPS=1M​∑m=1Md⁡\(𝐱m,𝐲\)−12​M​\(M−1\)​∑i=1M∑j=1j≠iMd⁡\(𝐱i,𝐱j\)\.\\mathrm\{TrackCRPS\}=\\frac\{1\}\{M\}\\sum\_\{m=1\}^\{M\}d\(\\mathbf\{x\}\_\{m\},\\mathbf\{y\}\)\-\\frac\{1\}\{2M\(M\-1\)\}\\sum\_\{i=1\}^\{M\}\\sum\_\{\\begin\{subarray\}\{c\}j=1\\\\ j\\neq i\\end\{subarray\}\}^\{M\}d\(\\mathbf\{x\}\_\{i\},\\mathbf\{x\}\_\{j\}\)\.\(31\)
![Refer to caption](https://arxiv.org/html/2609.38632v1/hurricane_michael_4panel.png)Figure 21:Hurricane Michael track forecasts from 50\-member MOSAIC \(top\) and U\-Cast \(bottom\) ensembles, comparing CRPS \(blue, left\) and DDM \(purple, right\)\. Michael occurred in 2018, within the validation period, and is included as an illustrative case study\.Table 17:Tropical\-cyclone ensemble\-mean track error and track CRPS \(km\)\.Hurricane IanHurricane MichaelTrack Error↓\\downarrowTrack CRPS↓\\downarrowTrack Error↓\\downarrowTrack CRPS↓\\downarrowMethodObjective1d3d5d1d3d5d1d2d3d4d1d2d3d4dMOSAICCRPS7389189516812542811562313054110153MOSAICDDM38445526396325176813817174786U\-CastCRPS3959245304314836751503082551112231U\-CastDDM353721024381233562107210244072141

Table 18:Number and percentage of ensemble members within 150, 200, and 300 km of the time\-matched IBTrACS center for Hurricane Ian\. MOSAIC is evaluated at 48 h and U\-Cast at 120 h, corresponding to the inset lead times as shown in Figures[22](https://arxiv.org/html/2609.38632#A4.F22)–[24](https://arxiv.org/html/2609.38632#A4.F24)\.![Refer to caption](https://arxiv.org/html/2609.38632v1/hurricane_ian_4panel_insets_150.png)Figure 22:50\-member ensemble track forecasts for Hurricane Ian initialized at 12 UTC on 23 September 2022, from MOSAIC \(top\) and U\-Cast \(bottom\), comparing CRPS \(blue, left\) and DDM \(purple, right\)\. Insets show the 48 h MOSAIC and 120 h U\-Cast forecast positions\. Dashed circles mark a 150 km radius around the time\-matched IBTrACS center\.![Refer to caption](https://arxiv.org/html/2609.38632v1/hurricane_ian_4panel_insets_200.png)Figure 23:Same as Figure[22](https://arxiv.org/html/2609.38632#A4.F22), but with a 200 km radius\.![Refer to caption](https://arxiv.org/html/2609.38632v1/hurricane_ian_4panel_insets_300.png)Figure 24:Same as Figure[22](https://arxiv.org/html/2609.38632#A4.F22), but with a 300 km radius\.