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Koopman Model Dimension Reduction via Variational Bayesian Inference and Graph Search

Selin Ezgi Ozcan, Mustafa Mert Ankarali

TL;DR

This work presents a hierarchical probabilistic approach to Koopman model identification that treats dictionary elements as random variables with inclusion flags, estimated via variational Bayesian inference. A spike‑and‑slab prior promotes sparsity, and the resulting inclusion probabilities form a directed graph whose condensed SCC structure enables a graph‑theoretic dictionary reduction that preserves the input–output behavior. The method yields improved numerical conditioning and stable long‑term predictions across three case studies (Lorenz, USV, Wiener‑Hammerstein) while drastically reducing model size. The practical impact is a principled, scalable pathway to compact Koopman models that maintain performance for control and identification tasks, with guidance on priors and a recommended reduction‑then‑robustification workflow.

Abstract

Koopman operator recently gained increasing attention in the control systems community for its abilities to bridge linear and nonlinear systems. Data driven Koopman operator approximations have established themselves as key enablers for system identification and model predictive control. Nonetheless, such methods commonly entail a preselected definition of states in the function space leading to high dimensional overfitting models and degraded long term prediction performances. We address this problem by proposing a hierarchical probabilistic approach for the Koopman model identification problem. In our method, elements of the model are treated as random variables and the posterior estimates are found using variational Bayesian (VB) inference updates. Our model distinguishes from others in the integration of inclusion flags. By the help of the inclusion flags, we intuitively threshold the probability of each state in the model. We then propose a graph search based algorithm to reduce the preselected states of the Koopman model. We demonstrate that our reduction method overcomes numerical instabilities and the reduced models maintain performance in simulated and real experiments.

Koopman Model Dimension Reduction via Variational Bayesian Inference and Graph Search

TL;DR

This work presents a hierarchical probabilistic approach to Koopman model identification that treats dictionary elements as random variables with inclusion flags, estimated via variational Bayesian inference. A spike‑and‑slab prior promotes sparsity, and the resulting inclusion probabilities form a directed graph whose condensed SCC structure enables a graph‑theoretic dictionary reduction that preserves the input–output behavior. The method yields improved numerical conditioning and stable long‑term predictions across three case studies (Lorenz, USV, Wiener‑Hammerstein) while drastically reducing model size. The practical impact is a principled, scalable pathway to compact Koopman models that maintain performance for control and identification tasks, with guidance on priors and a recommended reduction‑then‑robustification workflow.

Abstract

Koopman operator recently gained increasing attention in the control systems community for its abilities to bridge linear and nonlinear systems. Data driven Koopman operator approximations have established themselves as key enablers for system identification and model predictive control. Nonetheless, such methods commonly entail a preselected definition of states in the function space leading to high dimensional overfitting models and degraded long term prediction performances. We address this problem by proposing a hierarchical probabilistic approach for the Koopman model identification problem. In our method, elements of the model are treated as random variables and the posterior estimates are found using variational Bayesian (VB) inference updates. Our model distinguishes from others in the integration of inclusion flags. By the help of the inclusion flags, we intuitively threshold the probability of each state in the model. We then propose a graph search based algorithm to reduce the preselected states of the Koopman model. We demonstrate that our reduction method overcomes numerical instabilities and the reduced models maintain performance in simulated and real experiments.
Paper Structure (20 sections, 45 equations, 9 figures, 1 table, 2 algorithms)

This paper contains 20 sections, 45 equations, 9 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Koopman Operator and Data Driven Approximations
  4. Variational Bayesian (VB) Inference
  5. Condensation of a Graph
  6. Problem Statement
  7. Method
  8. VB inference of posterior distributions
  9. Update of $q(\rho)$
  10. Update of $q(\alpha_i)$
  11. Update of $q(\pi_i)$
  12. Update of $q(\beta_i)$
  13. Update of $q(\gamma_i)$
  14. Dictionary Reduction Using The Inclusion Matrix
  15. Results
  16. ...and 5 more sections

Figures (9)

  • Figure 1: The hierarchy of the random variables in the proposed model.
  • Figure 2: Reduced dictionary sizes of the Lorenz system with different $\epsilon$ across all measurement noise levels.
  • Figure 3: NMSE performance of full size models identified with different methods for the $y$ state of the Lorenz attractor in both training and test datasets.
  • Figure 4: NMSE performance of reduced size models identified with different methods for the $y$ state of the Lorenz attractor in both training and test datasets.
  • Figure 5: NMSE performance of full size models identified with different methods for the surge velocity state of USV in both training and test datasets.
  • ...and 4 more figures