Data-driven Discovery of Invariant Measures

Abstract

Invariant measures encode the long-time behaviour of a dynamical system. In this work, we propose an optimization-based method to discover invariant measures directly from data gathered from a system. Our method does not require an explicit model for the dynamics and allows one to target specific invariant measures, such as physical and ergodic measures. Moreover, it applies to both deterministic and stochastic dynamics in either continuous or discrete time. We provide convergence results and illustrate the performance of our method on data from the logistic map and a stochastic double-well system, for which invariant measures can be found by other means. We then use our method to approximate the physical measure of the chaotic attractor of the Rössler system, and we extract unstable periodic orbits embedded in this attractor by identifying discrete-time periodic points of a suitably defined Poincaré map. This final example is truly data-driven and shows that our method can significantly outperform previous approaches based on model identification.

Fix small mistake in analysis, extended examples (incl. comparison with alternative methods)

References in corpus (6)

• Data-driven approximation of the Koopman generator: Model reduction, system identification, and control
• Poincaré Maps for Multiscale Physics Discovery and Nonlinear Floquet Theory
• Data-Driven Stabilization of Periodic Orbits
• Deep Learning of Conjugate Mappings
• Auxiliary Functions as Koopman Observables: Data-Driven Analysis of Dynamical Systems via Polynomial Optimization
• Finding extremal periodic orbits with polynomial optimisation, with application to a nine-mode model of shear flow