# Generative machine learning methods for multivariate ensemble post-processing

arXiv: https://arxiv.org/abs/2211.01345 · HTML: https://mlsanity.com/paper/2211.01345/

## Key facts

- Cited by 26 papers in the map (data through 14 Oct 2026).
- Builds on 29 of its 88 references that are in the map.
- First posted on arXiv 26 Sep 2022.

## Builds on (most cited first, 29 of 29)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [Improved Techniques for Training Score-Based Generative Models](https://mlsanity.com/paper/2006.09011/) (2020)
- [InfoGAN: Interpretable Representation Learning by Information Maximizing Generative Adversarial Nets](https://mlsanity.com/paper/1606.03657/) (2016)
- [Skillful Precipitation Nowcasting using Deep Generative Models of Radar](https://mlsanity.com/paper/2104.00954/) (2021)
- [Generative Moment Matching Networks](https://mlsanity.com/paper/1502.02761/) (2015)
- [A Review on Generative Adversarial Networks: Algorithms, Theory, and Applications](https://mlsanity.com/paper/2001.06937/) (2020)
- [Training generative neural networks via Maximum Mean Discrepancy optimization](https://mlsanity.com/paper/1505.03906/) (2015)
- [Learning in Implicit Generative Models](https://mlsanity.com/paper/1610.03483/) (2016)
- [Neural networks for post-processing ensemble weather forecasts](https://mlsanity.com/paper/1805.09091/) (2018)
- [Stochastic Super-Resolution for Downscaling Time-Evolving Atmospheric Fields with a Generative Adversarial Network](https://mlsanity.com/paper/2005.10374/) (2020)
- [A Generative Deep Learning Approach to Stochastic Downscaling of Precipitation Forecasts](https://mlsanity.com/paper/2204.02028/) (2022)
- [Probabilistic Forecasting with Generative Networks via Scoring Rule Minimization](https://mlsanity.com/paper/2112.08217/) (2021)
- [Increasing the accuracy and resolution of precipitation forecasts using deep generative models](https://mlsanity.com/paper/2203.12297/) (2022)
- [Machine learning methods for postprocessing ensemble forecasts of wind gusts: A systematic comparison](https://mlsanity.com/paper/2106.09512/) (2021)
- [Physically Constrained Generative Adversarial Networks for Improving Precipitation Fields from Earth System Models](https://mlsanity.com/paper/2209.07568/) (2022)
- [Uncertainty Quantification in Complex Simulation Models Using Ensemble Copula Coupling](https://mlsanity.com/paper/1302.7149/) (2013)
- [Forecaster's Dilemma: Extreme Events and Forecast Evaluation](https://mlsanity.com/paper/1512.09244/) (2015)
- [Implicit Generative Copulas](https://mlsanity.com/paper/2109.14567/) (2021)
- [Evaluating forecasts for high-impact events using transformed kernel scores](https://mlsanity.com/paper/2202.12732/) (2022)
- [Log-normal distribution based EMOS models for probabilistic wind speed forecasting](https://mlsanity.com/paper/1407.3252/) (2014)
- [Multivariate Forecasting Evaluation: On Sensitive and Strictly Proper Scoring Rules](https://mlsanity.com/paper/1910.07325/) (2019)
- [Evaluating the Discrimination Ability of Proper Multivariate Scoring Rules](https://mlsanity.com/paper/2101.12693/) (2021)
- [Aggregating distribution forecasts from deep ensembles](https://mlsanity.com/paper/2204.02291/) (2022)
- [Convolutional autoencoders for spatially-informed ensemble post-processing](https://mlsanity.com/paper/2204.05102/) (2022)
- [Probabilistic multivariate electricity price forecasting using implicit generative ensemble post-processing](https://mlsanity.com/paper/2005.13417/) (2020)
- [Truncated generalized extreme value distribution based EMOS model for calibration of wind speed ensemble forecasts](https://mlsanity.com/paper/2008.11539/) (2020)
- [Joint probabilistic forecasting of wind speed and temperature using Bayesian model averaging](https://mlsanity.com/paper/1404.3681/) (2014)
- [Interpreting Generative Adversarial Networks for Interactive Image Generation](https://mlsanity.com/paper/2108.04896/) (2021)

## Built on by (most cited first, 26 of 26)

- [Uncertainty quantification for data-driven weather models](https://mlsanity.com/paper/2403.13458/) (2024)
- [Distributional Diffusion Models with Scoring Rules](https://mlsanity.com/paper/2502.02483/) (2025)
- [Probabilistic neural operators for functional uncertainty quantification](https://mlsanity.com/paper/2502.12902/) (2025)
- [Proper scoring rules for estimation and forecast evaluation](https://mlsanity.com/paper/2504.01781/) (2025)
- [EnScale: Temporally-consistent multivariate generative downscaling via proper scoring rules](https://mlsanity.com/paper/2509.26258/) (2025)
- [Postprocessing of Ensemble Weather Forecasts Using Permutation-invariant Neural Networks](https://mlsanity.com/paper/2309.04452/) (2023)
- [Postprocessing of point predictions for probabilistic forecasting of day-ahead electricity prices: The benefits of using isotonic distributional regression](https://mlsanity.com/paper/2404.02270/) (2024)
- [MVG-CRPS: A Robust Loss Function for Multivariate Probabilistic Forecasting](https://mlsanity.com/paper/2410.09133/) (2024)
- [Enforcing tail calibration when training probabilistic forecast models](https://mlsanity.com/paper/2506.13687/) (2025)
- [Graph Neural Networks and Spatial Information Learning for Post-Processing Ensemble Weather Forecasts](https://mlsanity.com/paper/2407.11050/) (2024)
- [A Gentle Introduction and Tutorial on Deep Generative Models in Transportation Research](https://mlsanity.com/paper/2410.07066/) (2024)
- [Generating ensembles of spatially-coherent in-situ forecasts using flow matching](https://mlsanity.com/paper/2504.03463/) (2025)
- [Generative Machine Learning for Multivariate Angular Simulation](https://mlsanity.com/paper/2504.21505/) (2025)
- [Probabilistic intraday electricity price forecasting using generative machine learning](https://mlsanity.com/paper/2506.00044/) (2025)
- [A Composite-Loss Graph Neural Network for the Multivariate Post-Processing of Ensemble Weather Forecasts](https://mlsanity.com/paper/2509.02784/) (2025)
- [Enhancing multivariate post-processed visibility predictions utilizing CAMS forecasts](https://mlsanity.com/paper/2406.14159/) (2024)
- [Improving probabilistic forecasts of extreme wind speeds by training statistical post-processing models with weighted scoring rules](https://mlsanity.com/paper/2407.15900/) (2024)
- [Improved probabilistic regression using diffusion models](https://mlsanity.com/paper/2510.04583/) (2025)
- [Uncertainty Quantification for Regression: A Unified Framework based on kernel scores](https://mlsanity.com/paper/2510.25599/) (2025)
- [CMIP X-MOS: Improving Climate Models with Extreme Model Output Statistics](https://mlsanity.com/paper/2311.03370/) (2023)
- [Machine learning-based probabilistic forecasting of solar irradiance in Chile](https://mlsanity.com/paper/2411.11073/) (2024)
- [Uncertainty Quantification of Wind Gust Predictions in the Northeast United States: An Evidential Neural Network and Explainable Artificial Intelligence Approach](https://mlsanity.com/paper/2502.00300/) (2025)
- [Learning low-dimensional representations of ensemble forecast fields using autoencoder-based methods](https://mlsanity.com/paper/2502.04409/) (2025)
- [Modeling Spatial Extremal Dependence of Precipitation Using Distributional Neural Networks](https://mlsanity.com/paper/2407.08668/) (2024)
- [Likelihood-Free Variational Autoencoders](https://mlsanity.com/paper/2504.17622/) (2025)
- [STIPP: Space-time in situ postprocessing over the French Alps using proper scoring rules](https://mlsanity.com/paper/2601.02882/) (2026)

## Abstract (arXiv)

Ensemble weather forecasts based on multiple runs of numerical weather prediction models typically show systematic errors and require post-processing to obtain reliable forecasts. Accurately modeling multivariate dependencies is crucial in many practical applications, and various approaches to multivariate post-processing have been proposed where ensemble predictions are first post-processed separately in each margin and multivariate dependencies are then restored via copulas. These two-step methods share common key limitations, in particular the difficulty to include additional predictors in modeling the dependencies. We propose a novel multivariate post-processing method based on generative machine learning to address these challenges. In this new class of nonparametric data-driven distributional regression models, samples from the multivariate forecast distribution are directly obtained as output of a generative neural network. The generative model is trained by optimizing a proper scoring rule which measures the discrepancy between the generated and observed data, conditional on exogenous input variables. Our method does not require parametric assumptions on univariate distributions or multivariate dependencies and allows for incorporating arbitrary predictors. In two case studies on multivariate temperature and wind speed forecasting at weather stations over Germany, our generative model shows significant improvements over state-of-the-art methods and particularly improves the representation of spatial dependencies.