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 ordinary differential equations (neural ODEs) that predict only state trajectories, UPN simultaneously models both state evolution and its associated uncertainty by parameterizing coupled differential equations for mean and covariance dynamics. The architecture is grounded in Gaussian moment closure approximation, which enables efficient analytical uncertainty propagation through nonlinear dynamics without requiring stochastic sampling or ensemble methods. UPN supports two operational modes: pure prediction from initial conditions, and adaptive filtering with sparse measurement updates when observations become available during the prediction horizon. The continuous-depth formulation provides principled uncertainty quantification in a single forward pass, handles irregularly-sampled observations naturally, and adapts evaluation strategy to each input’s complexity. Experimental results demonstrate UPN’s effectiveness across multiple domains: (1) four canonical non-chaotic dynamical systems achieve near-perfect 96.7 % confidence interval coverage with single-point Markovian initialization; (2) chaotic Lorenz attractor modeling maintains 94.5 % calibration while correctly capturing exponential uncertainty growth in a fully Markovian framework; (3) real-world CubeSat trajectory prediction achieves 89.6 % error reduction through integrated measurement updates; and (4) time-series forecasting on the ETTh1 benchmark dataset demonstrates 14 % improved accuracy and 6.6 × faster inference compared to Neural Stochastic Differential Equations (Neural SDEs). These gains stem from UPN’s analytical distribution evolution, which provides superior computational efficiency and calibration compared to sampling-based approaches.