Uncertainty Propagation Networks for Neural Ordinary Differential Equations
This addresses the need for principled uncertainty quantification in neural ODEs for applications such as time-series analysis and dynamical systems, representing a novel method for a known bottleneck.
The paper tackles the problem of incorporating uncertainty quantification into continuous-time modeling by introducing Uncertainty Propagation Networks (UPN), a family of neural differential equations that simultaneously model state evolution and its associated uncertainty, achieving effectiveness in domains like continuous normalizing flows, time-series forecasting, and trajectory prediction.
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.