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

TL;DR

This paper introduces a novel neural framework that directly evolves predictive probability distributions for dynamical systems, enabling more accurate and efficient uncertainty quantification without relying on ensembles.

Contribution

It develops a distribution-to-distribution neural forecasting method using kernel mean embeddings and mixture density networks, directly modeling the evolution of predictive distributions.

Findings

Captures distributional evolution in nonlinear dynamics

Produces skillful probabilistic forecasts without ensembles

Outperforms simplified benchmark models

Abstract

Probabilistic forecasting provides a principled framework for uncertainty quantification in dynamical systems by representing predictions as probability distributions rather than deterministic trajectories. However, existing forecasting approaches, whether physics-based or neural-network-based, remain fundamentally trajectory-oriented: predictive distributions are usually accessed through ensembles or sampling, rather than evolved directly as dynamical objects. A distribution-to-distribution (D2D) neural probabilistic forecasting framework is developed to operate directly on predictive distributions. The framework introduces a distributional encoding and decoding structure around a replaceable neural forecasting module, using kernel mean embeddings to represent input distributions and mixture density networks to parameterise output predictive distributions. This design enables recursive propagation of predictive uncertainty within a unified end-to-end neural architecture, with model training and evaluation carried out directly in terms of probabilistic forecast skill. The framework is demonstrated on the Lorenz63 chaotic dynamical system. Results show that the D2D model captures nontrivial distributional evolution under nonlinear dynamics, produces skillful probabilistic forecasts without explicit ensemble simulation, and remains competitive with, and in some cases outperforms, a simplified perfect model benchmark. These findings point to a new paradigm for probabilistic forecasting, in which predictive distributions are learned and evolved directly rather than reconstructed indirectly through ensemble-based uncertainty propagation.