Koopman Model Dimension Reduction via Variational Bayesian Inference and Graph Search
For researchers in control systems and system identification, this work addresses a practical bottleneck in Koopman operator approximations by providing a principled dimension reduction method.
The paper tackles the problem of overparameterized Koopman models that suffer from poor numerical conditioning and degraded long-term prediction. The proposed method uses variational Bayesian inference with inclusion flags and a graph search algorithm to reduce model dimension, improving numerical conditioning and preserving or improving prediction performance while substantially reducing dictionary size.
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, overparameterized models that may suffer from poor numerical conditioning and degraded long term prediction performance. 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 the proposed reduction improves numerical conditioning and can preserve or improve prediction performance while substantially reducing the dictionary size.