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.

翻译：本文提出了不确定性传播网络（UPN），这是一种新型的神经微分方程族，能够自然地将不确定性量化融入连续时间建模。与现有仅预测状态轨迹的神经ODE不同，UPN通过参数化均值和协方差动态的耦合微分方程，同时对状态演化及其相关不确定性进行建模。该架构通过求解状态与协方差演化的耦合ODE，在实现状态相关、可学习过程噪声的同时，高效地通过非线性动力学传播不确定性，避免了离散化伪影。这种连续深度架构能根据每个输入的复杂度自适应调整其评估策略，提供理论完备的不确定性量化，并自然地处理非均匀采样观测。实验结果表明，UPN在多个领域均表现出卓越性能：包括具备不确定性量化的连续归一化流（CNF）、具有良好校准置信区间的时间序列预测，以及在稳定和混沌动力系统中均能实现的鲁棒轨迹预测。