Uncertainty Propagation Networks for Neural Ordinary Differential Equations

Abstract

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.