# A Composite-Loss Graph Neural Network for the Multivariate Post-Processing of Ensemble Weather Forecasts

arXiv: https://arxiv.org/abs/2509.02784 · HTML: https://mlsanity.com/paper/2509.02784/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 41 references that are in the map.
- First posted on arXiv 2 Sep 2025.

## Builds on (most cited first, 16 of 16)

- [Inductive Representation Learning on Large Graphs](https://mlsanity.com/paper/1706.02216/) (2017)
- [Fast Graph Representation Learning with PyTorch Geometric](https://mlsanity.com/paper/1903.02428/) (2019)
- [Graph Neural Network for Traffic Forecasting: A Survey](https://mlsanity.com/paper/2101.11174/) (2021)
- [Neural networks for post-processing ensemble weather forecasts](https://mlsanity.com/paper/1805.09091/) (2018)
- [Generative machine learning methods for multivariate ensemble post-processing](https://mlsanity.com/paper/2211.01345/) (2022)
- [Uncertainty Quantification in Complex Simulation Models Using Ensemble Copula Coupling](https://mlsanity.com/paper/1302.7149/) (2013)
- [Assessing the calibration of high-dimensional ensemble forecasts using rank histograms](https://mlsanity.com/paper/1310.0236/) (2013)
- [Score-based calibration testing for multivariate forecast distributions](https://mlsanity.com/paper/2211.16362/) (2022)
- [Assessing the calibration of multivariate probabilistic forecasts](https://mlsanity.com/paper/2307.05846/) (2023)
- [A two-step machine learning approach to statistical post-processing of weather forecasts for power generation](https://mlsanity.com/paper/2207.07589/) (2022)
- [Graph Neural Networks and Spatial Information Learning for Post-Processing Ensemble Weather Forecasts](https://mlsanity.com/paper/2407.11050/) (2024)
- [Clustering-based spatial interpolation of parametric post-processing models](https://mlsanity.com/paper/2401.14393/) (2024)
- [Enhancing multivariate post-processed visibility predictions utilizing CAMS forecasts](https://mlsanity.com/paper/2406.14159/) (2024)
- [Distributional Regression U-Nets for the Postprocessing of Precipitation Ensemble Forecasts](https://mlsanity.com/paper/2407.02125/) (2024)
- [Graph Neural Networks for Enhancing Ensemble Forecasts of Extreme Rainfall](https://mlsanity.com/paper/2504.05471/) (2025)
- [Machine learning-based probabilistic forecasting of solar irradiance in Chile](https://mlsanity.com/paper/2411.11073/) (2024)

## Built on by (most cited first, 3 of 3)

- [On the sensitivity of machine-learned probabilistic weather forecast models to scale-aware scoring rules](https://mlsanity.com/paper/2607.19161/) (2026)
- [STIPP: Space-time in situ postprocessing over the French Alps using proper scoring rules](https://mlsanity.com/paper/2601.02882/) (2026)
- [Statistical versus machine learning-based spatial interpolation of post-processed ensemble weather forecasts](https://mlsanity.com/paper/2609.07512/) (2026)

## Abstract (arXiv)

Ensemble forecasting systems have advanced meteorology by providing probabilistic estimates of future states. Nonetheless, systematic biases often persist, making statistical post-processing essential. Traditional parametric post-processing techniques and machine learning-based methods can produce calibrated predictive distributions at specific locations and lead times, yet often struggle to capture dependencies across forecast dimensions. To address this, multivariate post-processing methods-such as ensemble copula coupling and the Schaake shuffle-are widely applied in a second step to restore realistic inter-variable or spatio-temporal dependencies. The aim of this study is the multivariate post-processing of ensemble forecasts using a graph neural network (dualGNN) trained with a composite loss function that combines the energy score (ES) and the variogram score (VS). The method is evaluated on two datasets: WRF-based solar irradiance forecasts over northern Chile and ECMWF visibility forecasts for Central Europe. The dualGNN consistently outperforms all empirical copula-based post-processed forecasts and shows significant improvements compared to graph neural networks trained solely on either the continuous ranked probability score or the ES, according to the evaluated multivariate verification metrics. Furthermore, for the WRF forecasts, the rank-order structure of the dualGNN forecasts captures valuable dependency information, enabling a more effective restoration of spatial relationships than either the raw numerical weather prediction ensemble or historical observational rank structures. Notably, incorporating VS into the loss function improved the univariate performance for both target variables compared to training on ES alone. Moreover, for the visibility forecasts, the ES-VS combination even outperformed the strongest calibrated reference in terms of univariate performance.