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

2026-03-26

AI Summary

This work proposes a novel distribution-to-distribution (D2D) prediction paradigm for probabilistic forecasting in dynamical systems, circumventing the limitations of conventional trajectory-based ensemble simulations that struggle to directly model the evolution of predictive distributions. The approach introduces an end-to-end neural architecture that represents input distributions via kernel mean embeddings, parameterizes output distributions using mixture density networks, and recursively propagates uncertainty through interchangeable neural modules—enabling direct learning of distributional dynamics without explicit ensemble simulation. Experiments on the Lorenz63 system demonstrate that the method accurately captures the evolution of probability distributions under nonlinear dynamics, yielding high-skill probabilistic forecasts that match or even surpass the performance of a reduced perfect-model benchmark.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Machine Learning: Calibration & Uncertainty Quantification
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks

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.

Problem

• probabilistic forecasting
• dynamical systems
• predictive distributions
• uncertainty quantification
• distribution-to-distribution

Innovation

• distribution-to-distribution
• neural probabilistic forecasting
• kernel mean embedding
• mixture density network
• uncertainty propagation