Data-driven system identification using quadratic embeddings of nonlinear dynamics

TL;DR

The paper introduces QENDy, a data-driven method that learns quadratic representations of nonlinear dynamical systems by embedding them into higher-dimensional spaces, enabling accurate system identification.

Contribution

QENDy is a novel approach that extends existing methods by embedding nonlinear dynamics into quadratic forms, improving identification accuracy and convergence analysis.

Findings

QENDy outperforms SINDy and deep learning in benchmark tests.

The method accurately identifies governing equations from trajectory data.

QENDy's convergence properties are analyzed and compared with SINDy.

Abstract

We propose a novel data-driven method called QENDy (Quadratic Embedding of Nonlinear Dynamics) that not only allows us to learn quadratic representations of highly nonlinear dynamical systems, but also to identify the governing equations. The approach is based on an embedding of the system into a higher-dimensional feature space in which the dynamics become quadratic. Just like SINDy (Sparse Identification of Nonlinear Dynamics), our method requires trajectory data, time derivatives for the training data points, which can also be estimated using finite difference approximations, and a set of preselected basis functions, called dictionary. We illustrate the efficacy and accuracy of QENDy with the aid of various benchmark problems and compare its performance with SINDy and a deep learning method for identifying quadratic embeddings. Furthermore, we analyze the convergence of QENDy and SINDy in the infinite data limit, highlight their similarities and main differences, and compare the quadratic embedding with linearization techniques based on the Koopman operator.