# A multi-scale loss formulation for learning a probabilistic model with proper score optimisation

arXiv: https://arxiv.org/abs/2506.10868 · HTML: https://mlsanity.com/paper/2506.10868/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 25 references that are in the map.
- First posted on arXiv 12 Jun 2025.

## Builds on (most cited first, 9 of 9)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [Neural General Circulation Models for Weather and Climate](https://mlsanity.com/paper/2311.07222/) (2023)
- [GenCast: Diffusion-based ensemble forecasting for medium-range weather](https://mlsanity.com/paper/2312.15796/) (2023)
- [Probabilistic Forecasting with Generative Networks via Scoring Rule Minimization](https://mlsanity.com/paper/2112.08217/) (2021)
- [Diffusion-LAM: Probabilistic Limited Area Weather Forecasting with Diffusion](https://mlsanity.com/paper/2502.07532/) (2025)
- [Stochastic Latent Transformer: Efficient Modelling of Stochastically Forced Zonal Jets](https://mlsanity.com/paper/2310.16741/) (2023)

## Built on by (most cited first, 6 of 6)

- [High-Resolution Probabilistic Data-Driven Weather Modeling with a Stretched-Grid](https://mlsanity.com/paper/2511.23043/) (2025)
- [WeatherNext 3: Increasing resolution and performance of global weather models with raw observations](https://mlsanity.com/paper/2609.03582/) (2026)
- [CSU-PCAST: A Dual-Branch Transformer Framework for medium-range ensemble Precipitation Forecasting](https://mlsanity.com/paper/2510.20769/) (2025)
- [On the sensitivity of machine-learned probabilistic weather forecast models to scale-aware scoring rules](https://mlsanity.com/paper/2607.19161/) (2026)
- [Revealing the Potential of Learnable Perturbation Ensemble Forecast Model for Tropical Cyclone Prediction](https://mlsanity.com/paper/2510.23794/) (2025)
- [Rescene: band-limited stochastic forcing turns a frozen neural weather operator into a climate emulator](https://mlsanity.com/paper/2608.09971/) (2026)

## Abstract (arXiv)

We assess the impact of a multi-scale loss formulation for training probabilistic machine-learned weather forecasting models. The multi-scale loss is tested in AIFS-CRPS, a machine-learned weather forecasting model developed at the European Centre for Medium-Range Weather Forecasts (ECMWF). AIFS-CRPS is trained by directly optimising the almost fair continuous ranked probability score (afCRPS). The multi-scale loss better constrains small scale variability without negatively impacting forecast skill. This opens up promising directions for future work in scale-aware model training.