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

The gist: DIAL introduces Dynamic and Informed Adjustment of the Native Stochastic Law, which controls a frozen model’s native stochasticity by learning from past forecasts to predict whether each variable and scale needs more or less spread, avoiding retraining or full backpropagation through the forecast

Problem Motivation

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 Reliability diagnostics reveal gaps in FourCastNet 3 (FCN3) (Bonev et al. 2025), NeuralGCM (Kochkov et al. 2024), and ArchesWeatherGen (AWG) (Couairon et al. T850 U850) Figure 1 shows that global averages can hide opposing reliability gaps across variables, spatial scales, forecast lead times, and weather states The patterns differ across systems, and FCN3 exhibits reliability gaps in both directions within a single model

DIAL Methodology

DIAL treats reliability repair as control of the native noise in a frozen weather model It uses signed ensemble constraints to state where spread should increase or decrease without assuming that the future weather distribution is known The core components include an Estimator that predicts reliability needs, a Pilot and recurrent memory to track forecast evolution, a Q-Critic that estimates how possible noise changes affect future skill and reliability, and a Controller that uses the Actuator’s variable- and scale-matched noise bases to make allowed updates within the remaining KL budget

Mathematical Formulation

DIAL formulates ensemble reliability repair as stochastic optimal control of a frozen model’s native noise under signed ensemble and KL constraints It seeks to improve multiscale probabilistic skill while satisfying ensemble-reliability and native-law fidelity constraints: min P ∈PA EP SMS(omegaeval, Y) s.t. J2nd(P) ≤ ϵrel, D¯path(P∥P0) ≤ βpath The ideal law is a Gibbs law tilted toward joint ensemble forecasts with lower combined forecast and reliability cost This ideal law is not directly executable, so DIAL approximates this tilt through a closed loop where the Estimator predicts reliability needs, and the Q-Critic estimates the action-conditioned cost-to-go

Control Mechanism

The Controller uses its gradient and the native KL geometry to choose a local noise update This is achieved by computing the unconstrained natural proposal a nat t = −G−1t qt, where Gt is the damped metric defined by λKLFKL(st) + γI The Controller then projects this proposal onto the feasible set Ft = Alegal ∩ BKL(b step t) using the damped Fisher metric to obtain a legal action a exec t

Evaluation and Results

DIAL is evaluated on FourCastNet 3 (FCN3) using later ERA5 years unseen during training Relative to the unmodified Native FCN3 reference, DIAL achieves an 8.27% improvement in the primary multiscale continuous ranked probability score (CRPS) DIAL provides better scale-resolved reliability than learned postprocessing and reward-based control baselines In Table 1, DIAL achieves an nMS-afCRPS of 0.9173, compared to the Native FCN3 score of 1.0000 The results show that DIAL realizes a PathKL of 0.0259 and gives J2nd/ϵrel = 0.591 under the registered ex-ante budget βpath = 0.030 Table S5 shows that increasing the budget to 0.045 only reduces J2nd/ϵrel while worsening both nMSafCRPS and variogram skill

Conclusion

DIAL adapts ensemble stochasticity during generation rather than calibrating completed forecasts It converts signed reliability gaps across scales into future costs and adjusts native noise through KL-natural updates within a Path-KL budget On frozen FCN3, DIAL improves multiscale skill and pooled reliability over learned postprocessing and rewardbased control at the registered budget Its local control is limited to the Actuator’s noise coordinates and requires a Pilot forecast and multi-member scoring

How it works

The Estimator predicts demand, not an action, and it runs once at initialization At step t, a deterministic mask removes leads that have already passed: rbt: = Mtrb The online state augments this fixed demand forecast with live information, forming st = (rbt:, Wt, t, aexec t−1, mt−1, brem t) The Q-Critic separates future multiscale score from lowrank moment contributions to produce the action-conditioned cost-to-go Qϕ,ξ(s, a)

Key Components and Constraints

DIAL learns only the two predictions required by the local solver: reliability demand from the Estimator and actionconditioned future cost from the Q-Critic The learning process involves supervising both heads with differences between paired branches isolating the action effect The Controller uses its gradient and the native KL geometry to obtain a legal action within the remaining KL budget

Final Summary

DIAL adapts ensemble stochasticity during generation rather than calibrating completed forecasts It converts signed reliability gaps across scales into future costs and adjusts native noise through KL-natural updates within a Path-KL budget On frozen FCN3, DIAL improves multiscale skill and pooled reliability over learned postprocessing and rewardbased control at the registered budget Its local control is limited to the Actuator’s noise coordinates and requires a Pilot forecast and multi-member scoring

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Training methods learn stochastic behavior together with the forecast model, but this behavior is usually fixed after deployment and cannot adapt to the current forecast state (Price et al. 2025; Lang et al. 2026; Bonev et al. 2025) Output calibration corrects completed forecasts, but cannot change how randomness enters the model or develops through weather dynamics (Chen et al. 2024; Ben Bouallègue et al. 2024) Inference guidance steers a pretrained generator during sampling without retraining it (Dhariwal and Nichol 2021; Ren et al.

1. Bold header: Closed-Loop Native-Noise Control

This system can control a frozen model’s native stochasticity by learning from past forecasts to predict whether each variable and scale needs more or less spread. It achieves this by combining predictions to make small noise updates at each step, avoiding the need for retraining the weather model.

2. Bold header: Reliability-Aware Stochastic Optimal Control

The system can formulate ensemble reliability repair as stochastic optimal control of a frozen model’s native noise under signed ensemble and KL constraints. It uses a closed loop where an Estimator predicts signed reliability demand and a Q-Critic estimates the action-conditioned cost-to-go, guiding the Controller to choose updates within the remaining KL budget.

3. Bold header: Path-KL Budget Enforcement

The system maintains a constraint via Path-KL budget which limits how far the controlled distribution can move from the model’s native distribution, defined as D¯path(P∥P0):= ν−1 KLDKL(P∥P0). This ensures that adjustments are feasible within the model's native geometry while satisfying reliability requirements.

4. Bold header: Signed Reliability Diagnostics

The system can identify material reliability gaps using the diagnostic cell metric C = −2 log SSRfinite-M, where Positive C means that error energy exceeds effective spread energy and therefore calls for more spread; negative C calls for less. This allows for a bidirectional response to underdispersed and overdispersed cells.

5. Bold header: Action-Conditioned Cost Estimation

The Q-Critic utilizes a pooled dual construction, producing the local cost-to-go as Qϕ,ξ(s, a) = Q skill ϕ (s, a) + (B⊤ r ξ)⊤Q h ϕ (s, a), allowing the Controller to use ∇aQϕ,ξ(st, 0) to choose updates based on both skill and reliability objectives.

6. Bold header: Native Fisher Geometry for Control

The system computes the native Fisher geometry using the second-order expansion of the KL divergence near zero action, defined by FKL(s) = ∇2aD¯KL,t(a s) a=0. This metric is used to propose a native proposal (a nat t = −G−1 t qt) which is then projected onto the feasible set.

7. Bold header: Forecast-Visible State Augmentation

The system operates with a state vector 'st' that augments fixed demand forecasts with live information, including the Pilot weather token and recurrent memory, enabling it to make decisions based on evolving forecast conditions without requiring verifying weather data for inference.