Learning dynamical systems from data: Gradient-based dictionary optimization

TL;DR

This paper introduces a gradient descent-based framework for learning optimal basis functions for Koopman operator approximation, enhancing data-driven analysis of dynamical systems across various benchmarks.

Contribution

It proposes a novel method to learn interpretable basis functions from data, improving upon fixed basis approaches in Koopman analysis.

Findings

Effective in diverse benchmark problems

Improves interpretability of basis functions

Compatible with EDMD, SINDy, PDE-FIND

Abstract

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.