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Uncertainty Propagation in Stochastic Systems via Mixture Models with Error Quantification

Eduardo Figueiredo, Andrea Patane, Morteza Lahijanian, Luca Laurenti

TL;DR

The paper addresses uncertainty propagation in discrete-time non-linear stochastic systems, where exact distributions are typically intractable. It proposes a stochastic-approximation framework that represents the system distribution at each time as a finite mixture $\hat{\mathbb{P}}_{x_t}$ of one-step kernels, with formal total variation ($TV$) distance guarantees to the true distribution $\mathbb{P}_{x_t}$ over a finite horizon $T$ and threshold $\delta$. The core methods include a partition-based propagation scheme, explicit $TV$ bounds that are computable in closed form for Gaussian noise, and an adaptive grid refinement algorithm that concentrates components in high-mass regions to minimize the bound. Theoretical results show convergence of the approximation as the partition becomes finer, and numerical experiments on control benchmarks (including a Dubin's car model) and chance-constrained planning demonstrate improved bound tightness versus uniform grids and practical applicability to safety-critical tasks. Overall, the framework enables tractable, provably correct uncertainty propagation with quantifiable guarantees that support planning and certification under uncertainty.

Abstract

Uncertainty propagation in non-linear dynamical systems has become a key problem in various fields including control theory and machine learning. In this work we focus on discrete-time non-linear stochastic dynamical systems. We present a novel approach to approximate the distribution of the system over a given finite time horizon with a mixture of distributions. The key novelty of our approach is that it not only provides tractable approximations for the distribution of a non-linear stochastic system, but also comes with formal guarantees of correctness. In particular, we consider the total variation (TV) distance to quantify the distance between two distributions and derive an upper bound on the TV between the distribution of the original system and the approximating mixture distribution derived with our framework. We show that in various cases of interest, including in the case of Gaussian noise, the resulting bound can be efficiently computed in closed form. This allows us to quantify the correctness of the approximation and to optimize the parameters of the resulting mixture distribution to minimize such distance. The effectiveness of our approach is illustrated on several benchmarks from the control community.

Uncertainty Propagation in Stochastic Systems via Mixture Models with Error Quantification

TL;DR

The paper addresses uncertainty propagation in discrete-time non-linear stochastic systems, where exact distributions are typically intractable. It proposes a stochastic-approximation framework that represents the system distribution at each time as a finite mixture of one-step kernels, with formal total variation () distance guarantees to the true distribution over a finite horizon and threshold . The core methods include a partition-based propagation scheme, explicit bounds that are computable in closed form for Gaussian noise, and an adaptive grid refinement algorithm that concentrates components in high-mass regions to minimize the bound. Theoretical results show convergence of the approximation as the partition becomes finer, and numerical experiments on control benchmarks (including a Dubin's car model) and chance-constrained planning demonstrate improved bound tightness versus uniform grids and practical applicability to safety-critical tasks. Overall, the framework enables tractable, provably correct uncertainty propagation with quantifiable guarantees that support planning and certification under uncertainty.

Abstract

Uncertainty propagation in non-linear dynamical systems has become a key problem in various fields including control theory and machine learning. In this work we focus on discrete-time non-linear stochastic dynamical systems. We present a novel approach to approximate the distribution of the system over a given finite time horizon with a mixture of distributions. The key novelty of our approach is that it not only provides tractable approximations for the distribution of a non-linear stochastic system, but also comes with formal guarantees of correctness. In particular, we consider the total variation (TV) distance to quantify the distance between two distributions and derive an upper bound on the TV between the distribution of the original system and the approximating mixture distribution derived with our framework. We show that in various cases of interest, including in the case of Gaussian noise, the resulting bound can be efficiently computed in closed form. This allows us to quantify the correctness of the approximation and to optimize the parameters of the resulting mixture distribution to minimize such distance. The effectiveness of our approach is illustrated on several benchmarks from the control community.
Paper Structure (21 sections, 3 theorems, 24 equations, 3 figures, 3 tables, 2 algorithms)

This paper contains 21 sections, 3 theorems, 24 equations, 3 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Notation
  4. System Description
  5. Problem Statement
  6. Approach
  7. Stochastic Approximation Scheme
  8. Total variation bounds
  9. Total Variation Bounds for the Gaussian Case
  10. Convergence Analysis
  11. Algorithm
  12. Refinement Procedure
  13. NUMERICAL EXPERIMENTAL RESULTS
  14. Tightness of the Bounds
  15. Application to chance constrained certification
  16. ...and 6 more sections

Key Result

Theorem 1

Let function $s ( x, x^{(k)} )$ be defined as Then, for any $t>0$ it holds that where $\text{TV} \left( \mathbb{P}_{x_0}, \hat{\mathbb{P}}_{x_0} \right)=0$.

Figures (3)

  • Figure 1: Example of a partition used to propagate the distribution of the System presented in Example \ref{['ex:gaussian-noise-explanation']} with $\varepsilon \sim \mathcal{N}(0, 0.1I)$. The initial grid (left) is refined (right) with a contribution threshold of $\gamma = 1 \times 10^{-3}$. We see that the algorithm decided to cut all regions except for the corners.
  • Figure 2: Upper row considers the Dubins car setting with low noise variance (i.e. $10^{-3}$ for the positions $x_1$ and $x_2$, and $10^{-4}$ for the steering angle $x_3$), while the lower row considers a higher variance noise ($10^{-2}$, and $2 \times 10^{-3}$, respectively). The mixtures are generated with only one refinement (see Algorithm \ref{['algo:refine-grid']}) with $\gamma = 10^{-6}$, resulting in mixtures of size 2500-3000. True distribution means Monte Carlo sampling from the system, while the plots on the right are generated by sampling our mixture distributions.
  • Figure 3: Bimodal setting for which we present the hitting probability study. The left image shows a Monte Carlo simulation of the system, while the right displays samples from our mixture approximation. $p_{\text{hit}}$ is computed w.r.t. the red set.

Theorems & Definitions (10)

  • Example 1
  • Definition 1: Total Variation
  • Example 2
  • Remark 1
  • Theorem 1
  • Corollary 1
  • Remark 2
  • Theorem 2
  • Remark 3
  • proof