# Postprocessing of Ensemble Weather Forecasts Using Permutation-invariant Neural Networks

arXiv: https://arxiv.org/abs/2309.04452 · HTML: https://mlsanity.com/paper/2309.04452/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 67 references that are in the map.
- First posted on arXiv 8 Sep 2023.

## Builds on (most cited first, 26 of 26)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [A Unified Approach to Interpreting Model Predictions](https://mlsanity.com/paper/1705.07874/) (2017)
- ["Why Should I Trust You?": Explaining the Predictions of Any Classifier](https://mlsanity.com/paper/1602.04938/) (2016)
- [Learning Important Features Through Propagating Activation Differences](https://mlsanity.com/paper/1704.02685/) (2017)
- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [A Survey Of Methods For Explaining Black Box Models](https://mlsanity.com/paper/1802.01933/) (2018)
- [FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2202.11214/) (2022)
- [A Survey on Neural Network Interpretability](https://mlsanity.com/paper/2012.14261/) (2020)
- [Order Matters: Sequence to sequence for sets](https://mlsanity.com/paper/1511.06391/) (2015)
- [A Survey on the Explainability of Supervised Machine Learning](https://mlsanity.com/paper/2011.07876/) (2020)
- [Towards a Neural Statistician](https://mlsanity.com/paper/1606.02185/) (2016)
- [Janossy Pooling: Learning Deep Permutation-Invariant Functions for Variable-Size Inputs](https://mlsanity.com/paper/1811.01900/) (2018)
- [Neural networks for post-processing ensemble weather forecasts](https://mlsanity.com/paper/1805.09091/) (2018)
- [On the Limitations of Representing Functions on Sets](https://mlsanity.com/paper/1901.09006/) (2019)
- [On the Benefits of Invariance in Neural Networks](https://mlsanity.com/paper/2005.00178/) (2020)
- [FSPool: Learning Set Representations with Featurewise Sort Pooling](https://mlsanity.com/paper/1906.02795/) (2019)
- [Deep Learning for Post-Processing Ensemble Weather Forecasts](https://mlsanity.com/paper/2005.08748/) (2020)
- [Model-agnostic Feature Importance and Effects with Dependent Features -- A Conditional Subgroup Approach](https://mlsanity.com/paper/2006.04628/) (2020)
- [Machine learning methods for postprocessing ensemble forecasts of wind gusts: A systematic comparison](https://mlsanity.com/paper/2106.09512/) (2021)
- [Universal approximations of permutation invariant/equivariant functions by deep neural networks](https://mlsanity.com/paper/1903.01939/) (2019)
- [Generative machine learning methods for multivariate ensemble post-processing](https://mlsanity.com/paper/2211.01345/) (2022)
- [On Deep Set Learning and the Choice of Aggregations](https://mlsanity.com/paper/1903.07348/) (2019)
- [Self-Attentive Ensemble Transformer: Representing Ensemble Interactions in Neural Networks for Earth System Models](https://mlsanity.com/paper/2106.13924/) (2021)
- [Aggregating distribution forecasts from deep ensembles](https://mlsanity.com/paper/2204.02291/) (2022)
- [Beyond Ensemble Averages: Leveraging Climate Model Ensembles for Subseasonal Forecasting](https://mlsanity.com/paper/2211.15856/) (2022)
- [Ensemble weather forecast post-processing with a flexible probabilistic neural network approach](https://mlsanity.com/paper/2303.17610/) (2023)

## Built on by (most cited first, 7 of 7)

- [Learning Enhanced Ensemble Filters](https://mlsanity.com/paper/2504.17836/) (2025)
- [Graph Neural Networks and Spatial Information Learning for Post-Processing Ensemble Weather Forecasts](https://mlsanity.com/paper/2407.11050/) (2024)
- [Valid Error Bars for Neural Weather Models using Conformal Prediction](https://mlsanity.com/paper/2406.14483/) (2024)
- [Self-attentive Transformer for Fast and Accurate Postprocessing of Temperature and Wind Speed Forecasts](https://mlsanity.com/paper/2412.13957/) (2024)
- [Learning low-dimensional representations of ensemble forecast fields using autoencoder-based methods](https://mlsanity.com/paper/2502.04409/) (2025)
- [Attention Kernels for Learning Maps Between Heavy-Tailed Measures](https://mlsanity.com/paper/2610.00564/) (2026)
- [TLCCSP: A Scalable Framework for Enhancing Time Series Forecasting with Time-Lagged Cross-Correlations](https://mlsanity.com/paper/2508.07016/) (2025)

## Abstract (arXiv)

Statistical postprocessing is used to translate ensembles of raw numerical weather forecasts into reliable probabilistic forecast distributions. In this study, we examine the use of permutation-invariant neural networks for this task. In contrast to previous approaches, which often operate on ensemble summary statistics and dismiss details of the ensemble distribution, we propose networks that treat forecast ensembles as a set of unordered member forecasts and learn link functions that are by design invariant to permutations of the member ordering. We evaluate the quality of the obtained forecast distributions in terms of calibration and sharpness and compare the models against classical and neural network-based benchmark methods. In case studies addressing the postprocessing of surface temperature and wind gust forecasts, we demonstrate state-of-the-art prediction quality. To deepen the understanding of the learned inference process, we further propose a permutation-based importance analysis for ensemble-valued predictors, which highlights specific aspects of the ensemble forecast that are considered important by the trained postprocessing models. Our results suggest that most of the relevant information is contained in a few ensemble-internal degrees of freedom, which may impact the design of future ensemble forecasting and postprocessing systems.