# A Distribution-to-Distribution Neural Probabilistic Forecasting Framework for Dynamical Systems

arXiv: https://arxiv.org/abs/2603.25370 · HTML: https://mlsanity.com/paper/2603.25370/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 48 references that are in the map.
- First posted on arXiv 26 Mar 2026.

## Builds on (most cited first, 12 of 12)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [Learning Phrase Representations using RNN Encoder-Decoder for Statistical Machine Translation](https://mlsanity.com/paper/1406.1078/) (2014)
- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [Kernel Mean Embedding of Distributions: A Review and Beyond](https://mlsanity.com/paper/1605.09522/) (2016)
- [Neural General Circulation Models for Weather and Climate](https://mlsanity.com/paper/2311.07222/) (2023)
- [Deep Learning for Day Forecasts from Sparse Observations](https://mlsanity.com/paper/2306.06079/) (2023)
- [SEEDS: Emulation of Weather Forecast Ensembles with Diffusion Models](https://mlsanity.com/paper/2306.14066/) (2023)
- [Building Energy Load Forecasting using Deep Neural Networks](https://mlsanity.com/paper/1610.09460/) (2016)
- [Beyond Strictly Proper Scoring Rules: The Importance of Being Local](https://mlsanity.com/paper/2012.12499/) (2020)
- [A Dictionary of Closed-Form Kernel Mean Embeddings](https://mlsanity.com/paper/2504.18830/) (2025)

## Abstract (arXiv)

Probabilistic forecasts provide a principled representation of forecast uncertainty through predictive distributions. Directly propagating predictive distributions with conventional physical models requires solving distributional evolution equations, a task well-known to be computationally prohibitive for high-dimensional nonlinear systems. Consequently, multi-step predictive distributions are commonly approximated using ensembles or sampled trajectories. A novel distribution-to-distribution (D2D) neural probabilistic forecasting framework is proposed with compatible distributional encoding and decoding interfaces around a replaceable neural forecasting module. Kernel mean embeddings represent input distributions, while mixture density networks parameterise outputs, enabling recursive multi-step forecasting in distributional form. The framework is demonstrated on the Lorenz63 system. Results show that the D2D model captures nontrivial distributional evolution, produces skillful probabilistic forecasts without explicit ensemble simulation, and remains competitive with a naively initialised perfect-model benchmark. This provides proof-of-concept evidence that predictive distributions can be recursively propagated through a neural forecasting model and evaluated in terms of probabilistic forecast performance.