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DIAL: Dynamic and Informed Adjustment of the Native Stochastic Law for Reliable Weather Ensembles

Yucheng Wang, Haonan Guo, Tianhao Zhao, Di Wang, Yu Wu, Zengmao Wang, Bo Du

arXiv:2610.21639v1cs.CV

TL;DR

Weather-ensemble reliability gaps can vary in direction across scales and lead times, while existing approaches have limited ability to adjust native randomness for joint ensemble reliability during a forecast. DIAL controls a frozen model’s native noise in a closed loop, reducing nMS-afCRPS by 8.27% from Native FCN3 and 1.57% from KL-PG.

  • Problem

    FCN3 reliability gaps can reverse direction across spatial scales and lead times, while inference guidance typically evaluates samples separately rather than updates’ effects on joint ensemble reliability.

    Source: Introduction · pp. 1–2

  • Method

    DIAL uses a closed loop to estimate reliability needs and guide local adjustments to a frozen weather model’s native noise without backpropagating through the full forecast.

    Source: Introduction · p. 2

  • Results

    8.27% lower nMS-afCRPS than Native FCN3 and 1.57% lower than KL-PG, the strongest baseline, at the validation-frozen budget.

    Source: Main Results · p. 7

  • Takeaways & Limitations

    DIAL provides an auditable reliability-aware stochastic closure without retraining the forecaster.

    Source: Conclusion · p. 8

    DIAL’s local control is limited to the Actuator’s noise coordinates and requires a Pilot forecast and multi-member scoring.

    Source: Conclusion · p. 8

Abstract

from arXiv · show

Ensemble weather forecasts can look reliable in global averages while hiding too little spread in some cases and too much in others. These errors vary across variables, spatial scales, lead times, and weather states. Adapting randomness during forecasting can correct these errors, but training fixes stochastic behavior beforehand and output calibration acts afterward. Inference guidance also acts during forecasting, but typically scores ensemble members separately rather than the ensemble as a whole. Reliability feedback arrives at the end, and tracing it backward through many chaotic forecast steps is expensive and unstable. We therefore introduce Dynamic and Informed Adjustment of the Native Stochastic Law (DIAL), which controls a frozen model's native stochasticity. DIAL learns from past forecasts to predict whether each variable and scale needs more or less spread. As a forecast develops, it estimates how possible noise changes will affect later forecast accuracy and reliability. It combines these predictions to make small noise updates at each step. This loop avoids retraining the weather model and full backpropagation through the forecast. We evaluate DIAL on FourCastNet 3 (FCN3) using later ERA5 years unseen during training. DIAL provides better scale-resolved reliability than learned postprocessing and reward-based control baselines. Relative to the unmodified Native FCN3 reference, DIAL achieves an 8.27% improvement in the primary multiscale continuous ranked probability score (CRPS).

Introduction

DIAL controls the native noise of a frozen weather model to correct signed, scale- and variable-specific ensemble reliability gaps. On later unseen ERA5 years, it improves the primary multiscale CRPS over Native FCN3.

Introduction

  • FCN3 has opposing reliability gaps within one model: it is overconfident at large scales and underconfident at the smallest scales, with some wind-variable gaps changing direction by lead time.Global averages can hide these signed gaps across variables, scales, leads, and weather states.Source: Introduction · p. 1, Figure 1 · p. 1
  • Training-time stochasticity is fixed after deployment, output calibration cannot alter randomness as it develops, and inference guidance generally scores samples separately rather than joint ensemble reliability.These limitations motivate closed-loop adjustment of a frozen model’s native noise as the forecast evolves.Source: Introduction · pp. 1–2
  • FCN3 injects noise every six hours, but reliability feedback arrives only after forecasts and verifying data are available, making full backpropagation expensive and vulnerable to exploding or vanishing gradients.The 28-step reliability objective is much longer than FCN3’s final 8-step training stage.Source: Introduction · p. 2
  • DIAL formulates reliability repair as stochastic optimal control of frozen-model native noise, using signed ensemble and KL constraints to approximate an ideal path law without full-forecast backpropagation.The Estimator and Pilot track reliability needs and forecast evolution; the Q-Critic estimates future effects, and the Controller guides variable- and scale-matched Actuator updates.Source: Introduction · p. 2
  • 8.27% improvement in primary multiscale CRPS over the unmodified Native FCN3 reference; DIAL is evaluated on later ERA5 years against learned postprocessing and reward-based control.The paper also reports that DIAL reduces reliability errors in both directions.Source: Introduction · p. 2

Related Work

DIAL differs from prior approaches by adjusting a frozen model’s native stochasticity during forecasting, targeting joint ensemble reliability rather than completed outputs or preferred individual samples.

Related Work

  • Probabilistic weather models use learned ensembles or stochastic perturbations (Lang et al. 2026; Bonev et al. 2025; Berner et al. 2017), but DIAL adjusts native noise during forecasting while keeping the model frozen.Their stochastic behavior is usually fixed before deployment, despite reliability varying with regime, scale, and lead time (Jung and Leutbecher 2008; Rodwell et al. 2018).Source: Related Work · p. 2
  • Post-processing calibrates completed ensembles and their dependence (Gneiting et al. 2005; Schefzik, Thorarinsdottir, and Gneiting 2013; Vannitsem et al. 2021; Chen et al. 2024; Ben Bouallègue et al. 2024), while DIAL acts inside the forecast.Post-processing cannot change where randomness enters a frozen model or how it develops through weather dynamics.Source: Related Work · p. 2
  • Generative guidance and reward fine-tuning steer sampling with likelihoods, rewards, gradients, or policies (Uehara et al. 2025; Ren et al. 2026; Clark et al. 2024; Black et al. 2024), whereas DIAL controls an ensemble law.Prior methods usually favor individual samples in one direction, while ensemble reliability is joint, scale-resolved, and bidirectional.Source: Related Work · p. 2
  • DIAL derives signed ensemble demand and applies budgeted updates in the model’s native KL geometry without full-forecast backpropagation.Reliability feedback is delayed, and long-horizon gradients are costly and unstable in chaotic dynamics (Mikhaeil, Monfared, and Durstewitz 2022); related stochastic-control methods use value, policy, or adjoint quantities, while KL regularization limits deviation from a reference process (Domingo-Enrich et al. 2025; Han, Razaviyayn, and Xu 2025; Blessing et al. 2025).Source: Related Work · p. 2

Method

The method frames weather forecasting around an initial state and a verifying sequence, and includes Reliability-Demand Learning and Signed Reliability Residuals.

Problem Formulation and DIAL Overview

PCA via SVD · FCN3 GT

DIAL adjusts a frozen weather model’s native stochastic law to improve multiscale forecast skill while constraining pooled ensemble reliability error and departure from the native law. It estimates reliability demand and future action costs from forecast-visible information, then applies feasible noise updates during forecasting.

FCN3 GT

  • DIAL seeks better multiscale probabilistic skill subject to pooled ensemble-reliability and normalized Path-KL constraints on departure from the frozen model’s native law.The reliability constraint compares ensemble moments with verifying weather across forecast cases; Path-KL uses no truth and decomposes into expected one-step kernel KL terms for budget tracking.Source: FCN3 GT · p. 3

From Pooled Reliability to an Ideal Stochastic Law

  • DIAL pools signed mean, second-moment, and predictability residuals across forecast initializations before applying a quadratic population-level reliability penalty.The residuals cover variable, spatial scale, and lead; the predictability row compares member–member and member–truth dependence.Source: From Pooled Reliability to an Ideal… · p. 4
  • The ideal Gibbs law tilts the native stochastic law toward joint ensemble paths with lower combined forecast and reliability cost, but its unknown future cost and normalizer prevent direct execution.DIAL estimates future cost from forecast-visible state and converts it into a legal local update without online truth.Source: From Pooled Reliability to an Ideal… · p. 4

Cost-to-Go from Forecast-Visible State

  • The Estimator predicts case-specific signed reliability demand from pre-forecast history, while the Q-Critic estimates action-conditioned cost-to-go using live Pilot information and recurrent memory.The Estimator runs once before forecasting; its lead-masked demand is not revised by memory or past actions.Source: Cost-to-Go from Forecast-Visible State · pp. 4–5

Projected KL-Natural Control of the Native Kernel

  • The Controller combines the Q-Critic gradient with native Fisher geometry, then projects its proposal onto legal actions and the remaining KL allowance.Because the Actuator-to-kernel map is nonlinear, DIAL recomputes exact KL and backtracks until both step and cumulative budgets hold.Source: Projected KL-Natural Control of the… · p. 5

Learning and Closed-Loop Inference

  • DIAL trains its Estimator and Q-Critic from verifying analyses and paired native-versus-adjusted branches, updates dual variables from pooled violations, then freezes learned components and control rules for inference.At inference, no verifying weather is available; the Estimator runs once and the Controller uses masked demand, Pilot state, memory, and the remaining KL ledger at each step.Source: Learning and Closed-Loop Inference · p. 5

Experiments

On later ERA5 test years, DIAL improves aggregate multiscale forecast skill over Native FCN3 and the strongest learned baseline while passing the registered reliability gate. Its gains are broad but not universal, and budget and ablation results show the value of live information and non-myopic, native-geometry control.

Experimental Setup

  • Experiments use frozen FCN3 forecasts on ERA5, with training and validation years preceding a sealed 2022 test set, eight fields, eight spectral bands, and 28 six-hour leads.The test set contains 60 initializations with 30 ensemble members each; dates are statistical units, and learned methods use three training seeds. Baselines include postprocessing methods and two generation-time controllers sharing DIAL’s state, actions, objective, and training budget; DRaFT-K-style differentiates through final K steps, while KL-PG omits DIAL’s action-conditioned Q-Critic and KL-natural solver.Source: Experimental Setup · pp. 6–7

Main Results

  • 8.27% lower nMS-afCRPS than Native FCN3 and 1.57% lower than KL-PG; the latter difference has a 95% CI of [0.05, 3.10]% and Holm-adjusted p = 0.022.DIAL has the highest fCRPSS and lowest RMS(C), while generative postprocessing gives the largest ensemble-mean RMSE improvement.Source: Main Results · p. 7, Table 1 · p. 7
  • DIAL is favored in 186 of 192 field–band–window cells, with six exceptions concentrated at the shortest spatial scales and two short-range T850 points not improving over KL-PG.Representative lead curves show the advantage over most of the seven-day forecast; the result does not establish universal cellwise dominance.Source: Main Results · p. 7
  • At βpath = 0.030, DIAL realizes Path-KL 0.0259 and J2nd/ϵrel = 0.591, passing the reliability gate with a 95% upper confidence bound of 0.79 while ensemble-mean RMSE remains better than Native.The bound is below one, satisfying the pooled-reliability criterion at the registered operating point.Source: Main Results · p. 7

Ablation Studies

  • DIAL beats both reward-control baselines on nMS-afCRPS at every nonzero budget and crosses the point threshold at a smaller budget, but extra budget beyond the registered point worsens nMS-afCRPS and variogram skill.Figure 4 varies the ex-ante budget while holding all components fixed; additional budget lowers J2nd.Source: Ablation Studies · p. 7
  • Replacing case-specific Estimator demand with training-only climatology or reusing the initial weather token at every step degrades nMS-afCRPS relative to full DIAL.Both variants retain the Controller, legal action space, and ex-ante budget; the results support both information sources without asserting an interaction.Source: Ablation Studies · p. 7, Table 2 · p. 7
  • A myopic cost degrades 102–168 h nMS skill, Euclidean updates use more Path-KL yet worsen nMS-afCRPS, and removing J2nd improves the primary score but raises the reliability ratio above one.These ablations support modeling effects on later forecasts and using native Fisher geometry; primary-score improvement alone does not meet pooled reliability.Source: Ablation Studies · p. 7, Table 2 · p. 7

Conclusion

DIAL adjusts a frozen forecaster’s native stochasticity during generation, improving multiscale skill and pooled reliability over learned postprocessing and reward-based control at the registered budget, though not every metric improves.

Conclusion

  • DIAL improves multiscale skill and pooled reliability over learned postprocessing and reward-based control at the registered budget, although not every metric improves.It adapts ensemble stochasticity during generation rather than calibrating completed forecasts.Source: Conclusion · p. 7
  • DIAL converts signed reliability gaps across scales into future costs and adjusts native noise through KL-natural updates within a Path-KL budget.Its local control is limited to the Actuator’s noise coordinates and requires a Pilot forecast and multi-member scoring, but provides an auditable reliability-aware stochastic closure without retraining the forecaster.Source: Conclusion · pp. 7–8

Scope and Reading Guide

The supplement provides derivations, protocol definitions, and extended diagnostics supporting the main paper’s central claims.

Formalization

  • Sections A–B formalize the pooled reliability objective, ideal path law, and forecast-visible approximation.Repeated definitions are included where needed to make proofs or tables independently readable.Source: Scope and Reading Guide · p. 9

Comparison protocol

Extended results

A Pooled Reliability and the Ideal Path Law · A.1 Augmented Joint-Path Setup and Assumptions

The setup treats scored forecast paths and an independent Pilot path as one augmented path variable, while only scored paths enter ensemble evaluation. Its ideal-path analysis assumes shared history and verifying-weather laws, admissible native-supported controls, integrability, and fixed reliability geometry.

A.1 Augmented Joint-Path Setup and Assumptions

  • The theoretical path combines the M scored evaluation paths with an independent Pilot path that informs the forecast but never enters ensemble scores.The native conditional law is P0(· | H), and legal Actuator updates define the reachable controlled-law family PA.Source: §A.1 · p. 9
  • Admissible policies are deterministic functions of forecast information, and scored and Pilot innovations are conditionally independent of verifying weather, which neither Pilot nor Controller observes at inference.Native and controlled systems share the law of (H, Y), and the verifying-weather kernel Q is common across admissible conditional path laws.Source: §A.1 · p. 9
  • Every considered controlled transition is absolutely continuous with respect to its native counterpart, so finite-Path-KL controlled laws satisfy P ≪ P0.Source: §A.1 · p. 9
  • The variational analysis assumes integrable multiscale scores and moment residuals, plus conditional exponential integrability for each considered dual vector and positive KL temperature.Under Equation (A.3), the exponentially tilted law belongs to the variational domain, which is not an additional restricted model family.Source: §A.1 · p. 9
  • The cell-weight matrix W is symmetric positive definite, and reliability projections, normalizers, cell weights, and KL normalization are fitted before evaluation and held fixed.Source: §A.1 · p. 9
  • The normalized native-law deviation measures departure only after legality is enforced and does not replace the Actuator’s stability, support, covariance, or balance constraints.The derivation uses the conditional chain rule for joint laws sharing the same history marginal.Source: §A.1 · p. 9

A.2 Finite-Member Scores and the Signed Spread–Error Gap … C.2 Method Boundaries

These sections formalize DIAL’s population-scoring target, fixed-dual and local-control approximations, and the forecast scope and method boundaries used for evaluation.

A.2 Finite-Member Scores and the Signed Spread–Error Gap

  • The primary α = 0.95 almost-fair score aggregates proper weighted marginal or projected scores, while signed C indicates whether spread should rise or fall without identifying the full joint law.Positive C means error energy exceeds effective spread energy; negative C calls for less spread.Source: §A.2 · pp. 9–10

A.3 Population Reliability Must Be Pooled Before Squaring

  • DIAL defines population reliability by pooling signed moment residuals over forecast and verifying-weather laws before squaring, avoiding casewise penalties for weather-to-weather variability.The casewise risk generally differs from the population moment constraint.Source: §A.3 · p. 10

A.4 Quadratic Dualization and the Gibbs Law; A.5 Path-KL Accounting and the Local Projection Identity

  • The fixed-ξ Gibbs law is a full-path skill-and-moment tilt, not a projection onto the zero-moment reliability set; augmented-process Path-KL sums conditional kernel KLs when initial laws coincide.The practical controller is a receding local approximation, and shared actions and Pilot feedback must remain in the conditioning information.Source: §A.4 · pp. 10–11, §A.5 · p. 11

B.1 Demand Is Predicted Once; Feedback Updates the Control State; B.2 Action-Conditioned Future Cost

  • The Q-Critic separates future multiscale score from low-rank moment contributions; for each head ℓ in {skill, h}, it enforces zero advantage at the native action.Source: §B.2 · p. 11

B.3 KL-Natural Proposal and Exact Feasibility

  • DIAL proposes local actions using a KL-natural update, then applies the legal actuator map and exact-KL backtracking or falls back to the native action.The exact executed-kernel sum enforces the remaining Path-KL ledger; the nonlinear executed action need not equal the quadratic metric projection.Source: §B.3 · p. 12

B.4 One Matched-Basis Actuator Library

  • The evaluated Actuator uses fixed source-matched bases with independent innovation-scale and persistence coordinates, while configurational bases remain unevaluated extensions.Hard feasibility precedes Path-KL fidelity accounting; weak response from a frozen basis does not establish that a trained matched-basis closure cannot work.Source: §B.4 · p. 12

B.5 When Live Feedback Can Improve on Stale Weather

  • Live-information policies cannot have higher optimal expected cost than stale-information policies under the same actions and constraints; strict improvement requires additional attainment and uniqueness conditions.The proposition is conditional, motivating checks that preferred actions vary with live state and that the Pilot token predicts this variation beyond stale information.Source: §B.5 · pp. 12–13

C.1 Forecast Scope and Frozen Evaluation Grid; C.2 Method Boundaries

  • Evaluation covers eight ERA5-verified fields across eight spectral bands and 28 six-hour leads, with low/high flow-index strata; comparisons distinguish output-only methods from native-noise generation-time controllers.Learned components and evaluation rules are frozen, and the independent-stream Pilot is excluded from ensemble scores; Path-KL applies only to generation-time controllers.Source: §C.1 · p. 13, §C.2 · p. 13

C.3 Metrics and Aggregation Orders … E.1 Bidirectional Response on Material Gaps

Across the extended comparisons, DIAL leads on the primary reliability measures while its spread response reverses direction between materially underdispersed and overdispersed cells. The analyses distinguish aggregation views and report point comparisons rather than inferential claims.

C.3 Metrics and Aggregation Orders

  • The field–lead curves and scorecard cells use different aggregation orders, so their results answer different questions and need not match after marginalization.Primary-weighted long-window mechanism skill retains registered field/band weights, while equal-cell Actuator diagnostics isolate where a coordinate restriction acts; they are not a second estimate of the primary endpoint.Source: §C.3 · p. 13
  • Signed-response analyses define material underdispersion as Cnative ≥0.15 and overdispersion as Cnative ≤−0.15; action alignment is the fraction with Cnative δs > 0.State and lead sign-change subsets are defined before method comparison, and the document reports point comparisons only; inferential claims would require paired forecast-date resampling and a pre-specified multiplicity procedure.Source: §C.3 · p. 13

C.4 Joint Consistency Rules

  • Cell-derived aggregates and diagnostics come from one field–band–lead–state tensor, while registered main-paper and ablation anchors retain their frozen primary aggregation.RMSE and J2nd/ϵrel are recomputed from the same error amplitudes and residual field with its registered auxiliary moment contribution; existing main-paper values are inherited without alteration.Source: §C.4 · p. 13

D.1 Aggregate Point Comparison with Static-SP

  • DIAL has the lowest nMS-afCRPS, RMS(C), and J2nd/ϵrel, and uses less Path-KL than both generation-time comparators at the registered budget.Generative postprocessing retains the largest ensemble-mean RMSE gain and strongest fixed-pair variogram skill (2.681%, versus 0.728% for DIAL); Static-SP improves the Native score but remains above the reliability threshold.Source: Table S1 · p. 14, §D.1 · p. 14

E.1 Bidirectional Response on Material Gaps

  • DIAL’s mean spread response is +0.1059 on materially underdispersed cells and −0.1809 on materially overdispersed cells, with lower residual RMS(C) than KL-PG and output-side comparators in both groups.The two groups are scored separately, so improvement cannot result from cancelling opposing signed residuals; they contain 1464 and 1267 field–band–lead–state cells, respectively.Source: §E.1 · p. 14, Table S2 · p. 14

D.2 Complete Eight-Field Lead Curves

  • DIAL has the highest lead-averaged skill in all eight fields, although its margin varies by lead; Z500 dips at short range before widening at medium and long leads.U850 and V850 show additional lead structure, while Z850, T500, U500, and V500 vary more smoothly; signed audits retain local DIAL losses.Source: §D.2 · p. 14, Figure S1 · p. 15

E.2 Weather-State and Lead Sign Changes … F.3 Forecast-Visible Information and Memory

Across weather-state and lead sign changes, DIAL maintains aligned spread adjustments and improves its primary-score margin; the audits also identify operating-point, actuator, and information restrictions that shape performance. Selected cases show both successful directional responses and retained local failures.

E.2 Weather-State and Lead Sign Changes

  • 3.256% is DIAL’s count-weighted primary-score margin across sign-changing groups, versus 1.377% on stable-sign cells; its action alignment stays above 0.93 in all three changing groups.The margin over the best static or output-only comparator rises to 2.945–3.555% on sign-changing subsets.Source: Table S3 · p. 15, §E.2 · p. 15

E.3 Selected Signed Challenge Cells

  • The U500 pair requires increased spread at 48 h but suppressed spread at 132 h, while selected examples include eight intended-direction DIAL responses and two wrong-direction responses.Generative postprocessing has lower |C| in five of ten selected rows; the exhaustive partition in Table S3 supplies aggregate alignment.Source: §E.3 · p. 15, Table S4 · p. 16

F.1 Registered Budget and Sensitivity Grid

  • At βpath = 0.030, DIAL improves primary score, pooled reliability, and variogram audit together; increasing the budget to 0.045 worsens nMS-afCRPS and variogram skill.The validation-frozen operating point is retained at 0.030.Source: §F.1 · p. 16, Table S5 · p. 16

F.3 Forecast-Visible Information and Memory

  • Removing any forecast-visible information source or recurrent memory degrades primary score and long-lead skill; removing the Pilot or both sources causes substantially larger losses than removing memory.No Pilot retains case-specific demand but replaces weather tokens with their training mean; the first row also replaces demand with its training climatology.Source: §F.3 · p. 16

F.2 Restrictions of the Matched-Basis Actuator

  • The full evaluated source-matched actuator performs strongest overall, while amplitude-only control retains 97.9% of full DIAL’s small-scale skill but 77.5% of its large-scale skill.Persistence-only retains substantial long-lead skill but has lower small-scale and aggregate skill; coupling intensity and persistence recovers complementary benefits.Source: §F.2 · p. 16, Table S6 · p. 17

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