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Propagation of Uncertainty with the Koopman Operator

Simone Servadio, Giovanni Lavezzi, Christian Hofmann, Di Wu, Richard Linares

TL;DR

This work tackles nonlinear uncertainty propagation by introducing a Koopman-operator framework that propagates full probability density functions (PDFs) directly through the system dynamics. It develops both analytical (Galerkin) and numerical (EDMD) methods to compute a Koopman solution flow, then leverages eigenfunction decomposition to invert the map for backward propagation of PDFs. PDFs can be propagated either as observables or via the inverted KO map, with a logarithmic simplification and a least-squares reduction to maintain tractability and enable recursive updates. A Duffing oscillator example demonstrates accurate PDF propagation and shows convergence between Galerkin and EDMD, validated against Monte Carlo simulations, highlighting the approach’s potential for direct PDF-based filtering in nonlinear systems.

Abstract

This paper proposes a new method to propagate uncertainties undergoing nonlinear dynamics using the Koopman Operator (KO). Probability density functions are propagated directly using the Koopman approximation of the solution flow of the system, where the dynamics have been projected on a well-defined set of basis functions. The prediction technique is derived following both the analytical (Galerkin) and numerical (EDMD) derivation of the KO, and a least square reduction algorithm assures the recursivity of the proposed methodology.

Propagation of Uncertainty with the Koopman Operator

TL;DR

This work tackles nonlinear uncertainty propagation by introducing a Koopman-operator framework that propagates full probability density functions (PDFs) directly through the system dynamics. It develops both analytical (Galerkin) and numerical (EDMD) methods to compute a Koopman solution flow, then leverages eigenfunction decomposition to invert the map for backward propagation of PDFs. PDFs can be propagated either as observables or via the inverted KO map, with a logarithmic simplification and a least-squares reduction to maintain tractability and enable recursive updates. A Duffing oscillator example demonstrates accurate PDF propagation and shows convergence between Galerkin and EDMD, validated against Monte Carlo simulations, highlighting the approach’s potential for direct PDF-based filtering in nonlinear systems.

Abstract

This paper proposes a new method to propagate uncertainties undergoing nonlinear dynamics using the Koopman Operator (KO). Probability density functions are propagated directly using the Koopman approximation of the solution flow of the system, where the dynamics have been projected on a well-defined set of basis functions. The prediction technique is derived following both the analytical (Galerkin) and numerical (EDMD) derivation of the KO, and a least square reduction algorithm assures the recursivity of the proposed methodology.
Paper Structure (14 sections, 53 equations, 5 figures)

This paper contains 14 sections, 53 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Propagation of Uncertainty in Hamiltonian Systems
  3. Generating the Koopman Solution Flow
  4. The Analytical Approach
  5. The Numerical Approach
  6. The Convergence of the Approaches
  7. The KO map inversion
  8. The Koopman Propagation of PDFs
  9. The PDF propagation as an observable
  10. The PDF Propagation via Map Inversion
  11. The Logarithmic Simplification
  12. Recursivity and Least Square Reduction
  13. Numerical Example
  14. Conclusion

Figures (5)

  • Figure 1: Eigenvalue comparison between EDMD and Galerkin.
  • Figure 2: Solution accuracy comparison between EDMD and KO.
  • Figure 3: PDF KOSTPM propagation with Monte Carlo validation
  • Figure 4: PDF propagation accuracy comparison at $t_f$: KOSTPM vs Monte Carlo.
  • Figure 5: PDF KOSTPM propagation for multiple time steps.