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.

翻译：概率预测通过将预测表示为概率分布而非确定性轨迹，为动力系统的不确定性量化提供了规范框架。然而，现有预测方法（无论是基于物理模型还是神经网络）本质上仍以轨迹为导向：预测分布通常通过集成或采样获得，而非作为动态对象直接演化。本文提出了一种分布到分布（D2D）神经概率预测框架，可直接对预测分布进行操作。该框架围绕可替换的神经预测模块构建了分布编码与解码结构，利用核均值嵌入表示输入分布，并采用混合密度网络参数化输出预测分布。这一设计使得在统一端到端神经架构中实现预测不确定性的递归传播，模型训练与评估直接基于概率预测技能进行。以Lorenz63混沌动力系统为例的验证表明，D2D模型能够捕捉非线性动力学下的非平凡分布演化，无需显式集成模拟即可生成高技能概率预测，且与简化完美模型基准相比具有竞争力，在某些情况下性能更优。这些发现为概率预测开辟了新范式——预测分布可直接学习与演化，而非通过基于集成的不确定性传播间接重构。