# Explanation of Dynamic Physical Field Predictions using WassersteinGrad: Application to Autoregressive Weather Forecasting

arXiv: https://arxiv.org/abs/2604.22580 · HTML: https://mlsanity.com/paper/2604.22580/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 28 of its 52 references that are in the map.
- First posted on arXiv 24 Apr 2026.

## Builds on (most cited first, 28 of 28)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
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- [Deep Inside Convolutional Networks: Visualising Image Classification Models and Saliency Maps](https://mlsanity.com/paper/1312.6034/) (2013)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Explainable Artificial Intelligence (XAI): Concepts, Taxonomies, Opportunities and Challenges toward Responsible AI](https://mlsanity.com/paper/1910.10045/) (2019)
- [Learning Important Features Through Propagating Activation Differences](https://mlsanity.com/paper/1704.02685/) (2017)
- [Certified Adversarial Robustness via Randomized Smoothing](https://mlsanity.com/paper/1902.02918/) (2019)
- [SmoothGrad: removing noise by adding noise](https://mlsanity.com/paper/1706.03825/) (2017)
- [On the Robustness of Interpretability Methods](https://mlsanity.com/paper/1806.08049/) (2018)
- [Fast Computation of Wasserstein Barycenters](https://mlsanity.com/paper/1310.4375/) (2013)
- [The Lipschitz Constant of Self-Attention](https://mlsanity.com/paper/2006.04710/) (2020)
- [A mathematical perspective on Transformers](https://mlsanity.com/paper/2312.10794/) (2023)
- [The Shattered Gradients Problem: If resnets are the answer, then what is the question?](https://mlsanity.com/paper/1702.08591/) (2017)
- [UNETR++: Delving into Efficient and Accurate 3D Medical Image Segmentation](https://mlsanity.com/paper/2212.04497/) (2022)
- [A Consistent and Efficient Evaluation Strategy for Attribution Methods](https://mlsanity.com/paper/2202.00449/) (2022)
- [Skillful joint probabilistic weather forecasting from marginals](https://mlsanity.com/paper/2506.10772/) (2025)
- [Local Explanation Methods for Deep Neural Networks Lack Sensitivity to Parameter Values](https://mlsanity.com/paper/1810.03307/) (2018)
- [Gradient based Feature Attribution in Explainable AI: A Technical Review](https://mlsanity.com/paper/2403.10415/) (2024)
- [Fast Discrete Distribution Clustering Using Wasserstein Barycenter with Sparse Support](https://mlsanity.com/paper/1510.00012/) (2015)
- [The Mean-Field Dynamics of Transformers](https://mlsanity.com/paper/2512.01868/) (2025)
- [NoiseGrad: Enhancing Explanations by Introducing Stochasticity to Model Weights](https://mlsanity.com/paper/2106.10185/) (2021)
- [Finding the right XAI method -- A Guide for the Evaluation and Ranking of Explainable AI Methods in Climate Science](https://mlsanity.com/paper/2303.00652/) (2023)
- [A multiscale analysis of mean-field transformers in the moderate interaction regime](https://mlsanity.com/paper/2509.25040/) (2025)
- [Wasserstein Barycenter Model Ensembling](https://mlsanity.com/paper/1902.04999/) (2019)
- [AdaptGrad: Adaptive Sampling to Reduce Noise](https://mlsanity.com/paper/2410.07711/) (2024)
- [Wasserstein Distances Made Explainable: Insights Into Dataset Shifts and Transport Phenomena](https://mlsanity.com/paper/2505.06123/) (2025)
- [On Spectral Properties of Gradient-based Explanation Methods](https://mlsanity.com/paper/2508.10595/) (2025)

## Abstract (arXiv)

As the demand to integrate Artificial Intelligence into high-stakes environments continues to grow, explaining the reasoning behind neural-network predictions has shifted from a theoretical curiosity to a strict operational requirement. Our work is motivated by the explanations of autoregressive neural predictions on dynamic physical fields, as in weather forecasting. Gradient-based feature attribution methods are widely used to explain the predictions on such data, in particular due to their scalability to high-dimensional inputs. It is also interesting to remark that gradient-based techniques such as SmoothGrad are now standard on images to robustify the explanations using pointwise averages of the attribution maps obtained from several noised inputs. Our goal is to efficiently adapt this aggregation strategy to dynamic physical fields. To do so, our first contribution is to identify a fundamental failure mode when averaging perturbed attribution maps on dynamic physical fields: stochastic input perturbations do not induce stationary amplitude noise in attribution maps, but instead cause a geometric displacement of the attributions. Consequently, pointwise averaging blurs these spatially misaligned features. To tackle this issue, we introduce WassersteinGrad, which extracts a geometric consensus of perturbed attribution maps by computing their entropic Wasserstein barycenter. The results, obtained on regional weather data and a meteorologist-validated neural model, demonstrate promising explainability properties of WassersteinGrad over gradient-based baselines across both single-step and autoregressive forecasting settings.