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

Peter Amorese, Morteza Lahijanian

TL;DR

The paper tackles formal uncertainty propagation for nonlinear continuous-time systems with unknown dynamics by introducing a continuum-dynamics perspective that converts probability propagation into a tractable volume-evolution problem in a transformed unit space. It builds a Taylor-series approximation of the evolving control-mass volume $V_\Omega(t;\tau)$, with derivatives computed via a recursive $\Gamma_k$ operator, and proves conditions under which the expansion converges analytically. A Bernstein polynomial regressor is shown to satisfy the required convergence, boundary, analytic, and bounding properties, enabling computable upper bounds $\bar{P}_\tau$ on $P(\hat{\phi}(\mathbf{x}_0,\tau) \in R)$. Practically, the method pairs Taylor-based volume bounds with flowpipe-based reachability to produce accurate, formal certificates for probabilistic predictions, including rare-event regions, demonstrated on Van der Pol and cartpole systems. This approach offers a scalable route to model-based uncertainty quantification with provable guarantees for learned continuous-time dynamics.

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.

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

TL;DR

The paper tackles formal uncertainty propagation for nonlinear continuous-time systems with unknown dynamics by introducing a continuum-dynamics perspective that converts probability propagation into a tractable volume-evolution problem in a transformed unit space. It builds a Taylor-series approximation of the evolving control-mass volume , with derivatives computed via a recursive operator, and proves conditions under which the expansion converges analytically. A Bernstein polynomial regressor is shown to satisfy the required convergence, boundary, analytic, and bounding properties, enabling computable upper bounds on . Practically, the method pairs Taylor-based volume bounds with flowpipe-based reachability to produce accurate, formal certificates for probabilistic predictions, including rare-event regions, demonstrated on Van der Pol and cartpole systems. This approach offers a scalable route to model-based uncertainty quantification with provable guarantees for learned continuous-time dynamics.

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.
Paper Structure (21 sections, 6 theorems, 37 equations, 4 figures, 1 algorithm)

This paper contains 21 sections, 6 theorems, 37 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notation
  3. Problem Formulation
  4. Uncertainty Propagation
  5. Control Mass Perspective
  6. Volume Function Taylor Expansion
  7. Bounding the Volume Function
  8. Choosing a Regressor
  9. Condition \ref{['cond:cue']} (Convergent Universal Estimator)
  10. Condition \ref{['cond:bcs']} (Boundary Conditions)
  11. Condition \ref{['cond:analytic']} (Analytic Volume Function)
  12. Condition \ref{['cond:antideriv']} (Closed-form Antiderivative)
  13. Condition \ref{['cond:bound']} (Bound Convergence)
  14. Backward Set Propagation
  15. Experimental Results
  16. ...and 6 more sections

Key Result

Lemma 1

Let $\Gamma_{k=0}(\mathbf{u}) = 1$ and $\Gamma_{k}(\mathbf{u})$ be recursively defined as Then, the $k$-th time derivative of the volume function is

Figures (4)

  • Figure 1: Example evolution of a PDF subject to a non-linear $\hat{f}$. We wish to integrate $p(\mathbf{x}_\tau)$ (blue density) at time $t=\tau$ over $\Omega_\tau(\tau)=R$ (pink region). At time $t=0$, the density is simple, i.e., Gaussian in $\mathbb{R}^n$ or uniform in $\mathbb{U}^n$, but becomes complex as time progresses. At time $t=\tau$, $R$ is a simple rectangle, and the shape becomes more complex as time regresses. Since probability mass is a conserved quantity, the volume and density of $\Omega_\tau$ fluctuate such that its probability mass is constant throughout time.
  • Figure 2: 2D Van der Pol: probability and flowpipe evolution (top, a.*), and rare-event counterparts (bottom, b.*).
  • Figure 3: 4D Cartpole
  • Figure 4: Expansion Degree $m$ Ablation Study

Theorems & Definitions (9)

  • Definition 1: Convergent Universal Estimator
  • Lemma 1
  • Remark 1
  • Theorem 2
  • Theorem 3
  • Corollary 4
  • Theorem 5
  • Theorem 6
  • Remark 2