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Gaussian Processes with Noisy Regression Inputs for Dynamical Systems

Tobias M. Wolff, Victor G. Lopez, Matthias A. Müller

TL;DR

The paper addresses learning nonlinear dynamical systems from noisy trajectory data by extending Gaussian process regression to propagate regression-input noise through the gradient of the posterior mean, capturing correlations induced by trajectory structure. It develops scalar and multidimensional formulations, deriving adjusted covariance structures that include off-diagonal terms representing consecutive-sample dependencies, and demonstrates superior performance over standard GP and prior noisy-input methods on logistic growth and three complex dynamical systems. The approach yields improved accuracy in scenarios with substantial input and measurement noise, with a practical impact on GP-based controllers and state estimators for nonlinear dynamics. Overall, the work provides a principled framework for input-noise propagation in GP regression for dynamical systems and outlines avenues for further refinement and comparison with variational methods.

Abstract

This paper is centered around the approximation of dynamical systems by means of Gaussian processes. To this end, trajectories of such systems must be collected to be used as training data. The measurements of these trajectories are typically noisy, which implies that both the regression inputs and outputs are corrupted by noise. However, most of the literature considers only noise in the regression outputs. In this paper, we show how to account for the noise in the regression inputs in an extended Gaussian process framework to approximate scalar and multidimensional systems. We demonstrate the potential of our framework by comparing it to different state-of-the-art methods in several simulation examples.

Gaussian Processes with Noisy Regression Inputs for Dynamical Systems

TL;DR

The paper addresses learning nonlinear dynamical systems from noisy trajectory data by extending Gaussian process regression to propagate regression-input noise through the gradient of the posterior mean, capturing correlations induced by trajectory structure. It develops scalar and multidimensional formulations, deriving adjusted covariance structures that include off-diagonal terms representing consecutive-sample dependencies, and demonstrates superior performance over standard GP and prior noisy-input methods on logistic growth and three complex dynamical systems. The approach yields improved accuracy in scenarios with substantial input and measurement noise, with a practical impact on GP-based controllers and state estimators for nonlinear dynamics. Overall, the work provides a principled framework for input-noise propagation in GP regression for dynamical systems and outlines avenues for further refinement and comparison with variational methods.

Abstract

This paper is centered around the approximation of dynamical systems by means of Gaussian processes. To this end, trajectories of such systems must be collected to be used as training data. The measurements of these trajectories are typically noisy, which implies that both the regression inputs and outputs are corrupted by noise. However, most of the literature considers only noise in the regression outputs. In this paper, we show how to account for the noise in the regression inputs in an extended Gaussian process framework to approximate scalar and multidimensional systems. We demonstrate the potential of our framework by comparing it to different state-of-the-art methods in several simulation examples.
Paper Structure (12 sections, 28 equations, 2 figures)

This paper contains 12 sections, 28 equations, 2 figures.

Table of Contents

  1. INTRODUCTION
  2. PRELIMINARIES AND PROBLEM SETTING
  3. Exact GP regression
  4. GP regression with noisy training inputs
  5. Problem formulation
  6. SCALAR SYSTEMS
  7. Analysis of regression input noise
  8. Application to logistic growth example
  9. MULTIDIMENSIONAL SYSTEMS
  10. Analysis of regression input noise
  11. Application to batch reactor, two-link planar robot, and cart-pole system
  12. CONCLUSION

Figures (2)

  • Figure 1: Simulation results of example (\ref{['def:log:growth']}) considering two different process noise variances as indicated in the titles of the plots. We implement the here proposed extension (referred to as "CCS" standing for "covariance of consecutive samples"), a standard GP (called "ST") and the approach proposed by mchutchon2015nonlinear (called "NI" standing for "noisy inputs", which is the abbreviation given by the authors in mchutchon2015nonlinear to describe their framework). We report the MSE as defined in (\ref{['def:mse']}), respectively.
  • Figure 2: Simulation results for a batch reactor, a two-link planar robot, and a cart pole system. The figures show the performances of the here proposed framework (referred to as "CCS" standing for "covariance of consecutive samples"), the extension by mchutchon2015nonlinear (called "NI" standing for "noisy inputs", which is the abbreviation given by the authors in mchutchon2015nonlinear to describe their framework), and a standard GP (called "ST") for randomly sampled test data. We report the MSE as defined in (\ref{['def:mse']}).

Theorems & Definitions (1)

  • Remark 1