Learning dynamical systems from data: Gradient-based dictionary optimization
This addresses the challenge of selecting problem-dependent basis functions in data-driven dynamical system analysis, which often requires domain knowledge, by providing an automated optimization approach.
The authors tackled the problem of learning optimal basis functions for approximating the Koopman operator in dynamical systems from data, presenting a gradient-based optimization framework that works with existing methods like EDMD, SINDy, and PDE-FIND, and demonstrated its efficacy on benchmark problems including the Ornstein-Uhlenbeck process and protein-folding data.
The Koopman operator plays a crucial role in analyzing the global behavior of dynamical systems. Existing data-driven methods for approximating the Koopman operator or discovering the governing equations of the underlying system typically require a fixed set of basis functions, also called dictionary. The optimal choice of basis functions is highly problem-dependent and often requires domain knowledge. We present a novel gradient descent-based optimization framework for learning suitable and interpretable basis functions from data and show how it can be used in combination with EDMD, SINDy, and PDE-FIND. We illustrate the efficacy of the proposed approach with the aid of various benchmark problems such as the Ornstein-Uhlenbeck process, Chua's circuit, a nonlinear heat equation, as well as protein-folding data.