Learning Nonlinear Continuous-Time Systems for Formal Uncertainty Propagation and Probabilistic Evaluation

TL;DR

This paper presents a new method for propagating uncertainty through unknown nonlinear ODEs by using Taylor series approximations, enabling formal probabilistic evaluation and effective rare event prediction.

Contribution

It introduces a continuum dynamics framework for learning and uncertainty propagation in unknown nonlinear systems, with proven convergence and soundness.

Findings

Effective uncertainty propagation in nonlinear ODEs.

Improved rare event prediction accuracy.

Theoretical guarantees for the method's soundness.

Abstract

Nonlinear ordinary differential equations (ODEs) are powerful tools for modeling real-world dynamical systems. However, propagating initial state uncertainty through nonlinear dynamics, especially when the ODE is unknown and learned from data, remains a major challenge. This paper introduces a novel continuum dynamics perspective for model learning that enables formal uncertainty propagation by constructing Taylor series approximations of probabilistic events. We establish sufficient conditions for the soundness of the approach and prove its asymptotic convergence. Empirical results demonstrate the framework's effectiveness, particularly when predicting rare events.