Recovering nonlinear dynamics via Koopman Theory
In this talk, we will present the main building blocks that allow for an interpretative analysis and process of dynamical systems data. The overarching theme of our work is based on the theory of Bernard Koopman (1931). Key to our approach is the interplay between the underlying dynamics and its embedding onto an infinite-dimensional space of scalar functions. This embedding encodes a potentially nonlinear system via a linear object known as the Koopman Operator. Koopman's perspective is advantageous since it involves the manipulation of linear operators which can be done efficiently. Moreover, algebraic properties of the Koopman matrix are directly associated with dynamical features of the system, which in turn are linked to high-level questions. Overall, the combination of Koopman Theory with novel dimensionality reduction techniques and data science approaches leads to a highly powerful framework.
To demonstrate the effectiveness of our approach, we consider several challenging problems in various fields. In geometry processing, we show how optimizing for a spectral basis and a Koopman operator (a functional map) leads to improved shape matching results. In fluid mechanics, we develop a provably convergent ADMM scheme for computing Koopman operators that admits state-of-the-art results on data with high levels of sensor noise. In image processing, our methodology generates a discrete transform of a nonlinear flow as faster as two orders of magnitude when compared to existing approaches. Finally, we construct a novel framework for recovering dynamics from questionnaire data that arise in the social sciences.