Auxiliary Functions as Koopman Observables: Data-Driven Analysis of Dynamical Systems via Polynomial Optimization
This provides a flexible, data-driven tool for researchers in dynamical systems analysis, though it is incremental as it builds on established Koopman operator techniques.
The paper tackles the problem of analyzing dynamical systems without explicit model discovery by using auxiliary functions as Koopman observables, implemented via a semidefinite program that works for both deterministic and stochastic data, with rigorous convergence results demonstrated through examples like discovering Lyapunov functions and bounding extrema over attractors.
We present a flexible data-driven method for dynamical system analysis that does not require explicit model discovery. The method is rooted in well-established techniques for approximating the Koopman operator from data and is implemented as a semidefinite program that can be solved numerically. Furthermore, the method is agnostic of whether data is generated through a deterministic or stochastic process, so its implementation requires no prior adjustments by the user to accommodate these different scenarios. Rigorous convergence results justify the applicability of the method, while also extending and uniting similar results from across the literature. Examples on discovering Lyapunov functions, performing ergodic optimization, and bounding extrema over attractors for both deterministic and stochastic dynamics exemplify these convergence results and demonstrate the performance of the method.