Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Formal Uncertainty Propagation for Stochastic Dynamical Systems with Additive Noise

Steven Adams, Eduardo Figueiredo, Luca Laurenti

TL;DR

This work addresses the challenge of propagating uncertainty in discrete-time nonlinear stochastic systems with additive noise when the initial and noise distributions are themselves uncertain. It develops a tractable framework that over-approximates the evolving set of state distributions with $ ho$-Wasserstein balls centered at Gaussian-mixture approximations of the push-forward distributions, leveraging quantization and mixture-distribution properties. The authors derive computable radii guarantees that ensure containment of the true distributional evolution, and prove convergence of the radius for contracting dynamics via a fixed-point argument. Empirical results on benchmarks including nonlinear, switched, and neural-network–driven dynamics demonstrate the framework’s ability to produce informative, formally guaranteed uncertainty bounds and illustrate how hyperparameters affect the tightness of the bounds. This approach enables formal uncertainty quantification for complex stochastic systems and supports robust planning and control under distributional ambiguity.

Abstract

In this paper, we consider discrete-time non-linear stochastic dynamical systems with additive process noise in which both the initial state and noise distributions are uncertain. Our goal is to quantify how the uncertainty in these distributions is propagated by the system dynamics for possibly infinite time steps. In particular, we model the uncertainty over input and noise as ambiguity sets of probability distributions close in the $ρ$-Wasserstein distance and aim to quantify how these sets evolve over time. Our approach relies on results from quantization theory, optimal transport, and stochastic optimization to construct ambiguity sets of distributions centered at mixture of Gaussian distributions that are guaranteed to contain the true sets for both finite and infinite prediction time horizons. We empirically evaluate the effectiveness of our framework in various benchmarks from the control and machine learning literature, showing how our approach can efficiently and formally quantify the uncertainty in linear and non-linear stochastic dynamical systems.

Formal Uncertainty Propagation for Stochastic Dynamical Systems with Additive Noise

TL;DR

This work addresses the challenge of propagating uncertainty in discrete-time nonlinear stochastic systems with additive noise when the initial and noise distributions are themselves uncertain. It develops a tractable framework that over-approximates the evolving set of state distributions with -Wasserstein balls centered at Gaussian-mixture approximations of the push-forward distributions, leveraging quantization and mixture-distribution properties. The authors derive computable radii guarantees that ensure containment of the true distributional evolution, and prove convergence of the radius for contracting dynamics via a fixed-point argument. Empirical results on benchmarks including nonlinear, switched, and neural-network–driven dynamics demonstrate the framework’s ability to produce informative, formally guaranteed uncertainty bounds and illustrate how hyperparameters affect the tightness of the bounds. This approach enables formal uncertainty quantification for complex stochastic systems and supports robust planning and control under distributional ambiguity.

Abstract

In this paper, we consider discrete-time non-linear stochastic dynamical systems with additive process noise in which both the initial state and noise distributions are uncertain. Our goal is to quantify how the uncertainty in these distributions is propagated by the system dynamics for possibly infinite time steps. In particular, we model the uncertainty over input and noise as ambiguity sets of probability distributions close in the -Wasserstein distance and aim to quantify how these sets evolve over time. Our approach relies on results from quantization theory, optimal transport, and stochastic optimization to construct ambiguity sets of distributions centered at mixture of Gaussian distributions that are guaranteed to contain the true sets for both finite and infinite prediction time horizons. We empirically evaluate the effectiveness of our framework in various benchmarks from the control and machine learning literature, showing how our approach can efficiently and formally quantify the uncertainty in linear and non-linear stochastic dynamical systems.
Paper Structure (21 sections, 3 theorems, 18 equations, 2 figures, 2 tables, 1 algorithm)

This paper contains 21 sections, 3 theorems, 18 equations, 2 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notation
  3. Problem Formulation
  4. Problem Statement
  5. Approach
  6. Preliminaries
  7. Quantization of Probability Distributions
  8. Mixture Distributions
  9. Approximation Scheme
  10. Ambiguity Ball Centers via Mixture Distributions
  11. The Radii of the Ambiguity Balls
  12. Algorithm
  13. Experiments
  14. Qualitative analysis of the ambiguity set evolution
  15. Parametric analysis
  16. ...and 6 more sections

Key Result

Proposition 1

Let $\bar{\mathbb{P}}_{{{x}}_1}, \hdots,\bar{\mathbb{P}}_{{{x}}_K}$ be defined according to Eqn eq:approx_centers. For each $k\in\{0,\hdots,K-1\}$, iteratively define Then, $\forall k\in\mathbb{N}$ it holds that ${\mathbb{S}}_{{{x}}_{k}} \subseteq {\mathbb{B}}_{\theta_{{{x}}_k}}(\bar{\mathbb{P}}_{{{x}}_{k}}).$

Figures (2)

  • Figure 1: Schematic representation of our proposed approach for $k=1$. As ${\mathbb{S}}_{{{x}}_1}$ is generally intractable, we over-approximate it with a $\rho$-Wasserstein ambiguity ball centered at $\bar{\mathbb{P}}_{{{x}}_{1}} \approx (f\#\bar{\mathbb{P}}_{{{x}}_{0}}) \ast \bar{\mathbb{P}}_{{{\omega}}}$, that is, at the propagation of the center of the ambiguity ball at the previous time step through the system dynamics according to Eqn \ref{['eq:TrueDynPropagation']}. The error in this approximation is formally accounted for in the choice of $\theta_{{{x}}_1}$.
  • Figure 2: Vector fields (left column), samples from the true uncertainty sets ${\mathbb{S}}_{{{x}}_k}$ obtained via Monte Carlo sampling (center column), and samples from the centers $\bar{\mathbb{P}}_{{{x}}_k}$ of the ambiguity sets obtained via Algorithm \ref{['alg:propagate-ball']} (right column) for different benchmarks (by row), evaluated over 20 time-step. Sampled states are plotted as as dots, with the blue-red gradient indicating time evolution from the initial to the final time step. The circles, centered at the mean of $\bar{\mathbb{P}}_{{{x}}_k}$ with radius $\theta_{{{x}}_k}$, are guaranteed to contain the mean of the true distributions. The piecewise linear dynamics result from a switched system by the optimal switching strategy in gracia2025efficient, which steers the system toward the target region while avoiding the obstacles.

Theorems & Definitions (6)

  • Remark 1
  • Remark 2
  • Example 1
  • Proposition 1
  • Proposition 2
  • Proposition 3