Learning Dynamical Systems via Koopman Operator Regression in Reproducing Kernel Hilbert Spaces
This work addresses the challenge of data-driven modeling of dynamical systems for researchers in machine learning and control, but it is incremental as it builds on known Koopman operator methods.
The authors tackled the problem of learning Koopman operators for dynamical systems from finite data by formalizing a framework in reproducing kernel Hilbert spaces and proposing a reduced-rank regression estimator. They derived learning bounds and showed in experiments that this estimator improves forecasting and mode decomposition over existing methods.
We study a class of dynamical systems modelled as Markov chains that admit an invariant distribution via the corresponding transfer, or Koopman, operator. While data-driven algorithms to reconstruct such operators are well known, their relationship with statistical learning is largely unexplored. We formalize a framework to learn the Koopman operator from finite data trajectories of the dynamical system. We consider the restriction of this operator to a reproducing kernel Hilbert space and introduce a notion of risk, from which different estimators naturally arise. We link the risk with the estimation of the spectral decomposition of the Koopman operator. These observations motivate a reduced-rank operator regression (RRR) estimator. We derive learning bounds for the proposed estimator, holding both in i.i.d. and non i.i.d. settings, the latter in terms of mixing coefficients. Our results suggest RRR might be beneficial over other widely used estimators as confirmed in numerical experiments both for forecasting and mode decomposition.