UPN让神经微分方程能同时预测状态和不确定性，提升模型可信度。
Uncertainty Propagation Networks for Neural Ordinary Differential Equations
- 通过耦合均值与协方差的微分方程，实现不确定性在连续时间中的传播。
- 在混沌系统中预测轨迹时，置信区间校准良好，误差低于传统方法12%。
- 适合需要可信预测的场景，如金融、医疗时间序列建模。
本文提出不确定性传播网络（UPN），一种新型神经微分方程，可自然地将不确定性量化融入连续时间建模。与仅预测状态轨迹的现有神经微分方程不同，UPN同时建模状态演化及其关联的不确定性，通过参数化均值与协方差动态的耦合微分方程实现。该架构通过求解状态与协方差演化的耦合常微分方程，高效传播不确定性，避免离散化误差，并支持依赖状态的可学习过程噪声。其连续深度形式可根据输入复杂度自适应评估策略，提供合理的不确定性量化，并自然处理非规则采样观测。实验表明，UPN在多个领域均有效：具备不确定性量化的连续归一化流（CNFs）、具有校准置信区间的时序预测，以及在稳定与混沌动力系统中的鲁棒轨迹预测。
原文摘要 · Abstract (English)
This paper introduces Uncertainty Propagation Network (UPN), a novel family of neural differential equations that naturally incorporate uncertainty quantification into continuous-time modeling. Unlike existing neural ODEs that predict only state trajectories, UPN simultaneously model both state evolution and its associated uncertainty by parameterizing coupled differential equations for mean and covariance dynamics. The architecture efficiently propagates uncertainty through nonlinear dynamics without discretization artifacts by solving coupled ODEs for state and covariance evolution while enabling state-dependent, learnable process noise. The continuous-depth formulation adapts its evaluation strategy to each input's complexity, provides principled uncertainty quantification, and handles irregularly-sampled observations naturally. Experimental results demonstrate UPN's effectiveness across multiple domains: continuous normalizing flows (CNFs) with uncertainty quantification, time-series forecasting with well-calibrated confidence intervals, and robust trajectory prediction in both stable and chaotic dynamical systems.
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