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

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.

💡 Research Summary

The paper tackles the dual challenge of learning an unknown nonlinear continuous‑time dynamical system from data while providing formally guaranteed bounds on the probability that the system’s state will lie in a prescribed region at a future time. The authors first formalize the learning problem: given a dataset of state‑derivative pairs, they seek a “convergent universal estimator” – a parameterized model whose loss is convex and whose parameters, together with an ever‑growing dataset, converge in any (C^k) norm to the true vector field. This guarantees that any subsequent analysis performed on the learned model inherits the same asymptotic accuracy.

Uncertainty propagation is traditionally split into (i) propagating an initial probability density through the nonlinear flow and (ii) integrating the resulting density over the target set. Both steps are analytically intractable for general nonlinear ODEs. The authors introduce a continuum‑dynamics viewpoint: they treat the collection of infinitesimal particles as a “control mass” whose total probability mass is conserved. By mapping the initial Gaussian distribution to a uniform distribution on the unit hyper‑cube (U_n=(0,1)^n) via component‑wise cumulative distribution functions, the initial density becomes trivial, and the target region is also mapped to a hyper‑rectangle in the same space. In this transformed space the probability of interest reduces to the volume of the transformed control mass at time zero.

The key technical contribution is the construction of a scalar volume function (V_\Omega(t;\tau)) that records the volume of the control mass at intermediate time (t) for a fixed horizon (\tau). Using Reynolds transport theorem, the first time derivative of this volume is expressed as an integral of the divergence of the learned vector field over the current control mass. Higher‑order derivatives are obtained by repeatedly applying the theorem with appropriate scalar fields. The authors then approximate (V_\Omega(t;\tau)) by a Taylor series centered at (t=\tau), where the geometry of the control mass is simple (a hyper‑rectangle) and the required derivatives can be computed analytically for a class of models that satisfy boundary conditions (the flow vanishes on the hyper‑cube faces). Truncating the series at order (m) yields an explicit bound (\tilde V_m(0;\tau)) on the desired probability. The paper proves sufficient conditions under which the series converges to the true volume as (m\to\infty) and shows that the bound is sound (i.e., it never underestimates the true probability).

To make the approach practical, the authors design the learning architecture so that the divergence and its derivatives are readily available (e.g., by using neural networks with analytically tractable Jacobians and enforcing zero flow at the hyper‑cube boundaries). They also demonstrate how existing off‑the‑shelf reachability tools can be integrated to tighten the bound for longer horizons, effectively correcting the volume estimate when higher‑order terms become expensive to compute.

Empirical evaluation on synthetic benchmarks and real‑world dynamical systems (including a nonlinear oscillator and a robotic arm) confirms that the Taylor‑based bound is significantly tighter than naive Monte‑Carlo estimates, especially for rare events with probabilities on the order of (10^{-6}) or lower. The method achieves comparable accuracy with far fewer samples and provides a deterministic upper bound, which sampling methods lack. Moreover, the combination with reachability analysis extends the usable prediction horizon without sacrificing rigor.

In summary, the paper introduces a novel continuum‑dynamics framework that transforms probabilistic uncertainty propagation into a scalar volume estimation problem, solves it via a Taylor series expansion, and couples it with a specially constrained learning model and reachability tools. This yields formally provable, asymptotically convergent probability bounds for nonlinear continuous‑time systems, offering a powerful new tool for safety verification, risk assessment, and rare‑event prediction in cyber‑physical and control applications.