Generalized Quadratic Embeddings for Nonlinear Dynamics using Deep Learning
This work addresses the challenge of simplifying nonlinear dynamics modeling for engineering design, offering a method to apply common tools across various systems, though it appears incremental by building on the lifting principle with deep learning.
The authors tackled the problem of modeling nonlinear dynamical systems by proposing a data-driven method to identify coordinate transformations that enable representation as quadratic systems, achieving superior performance compared to existing techniques.
The engineering design process often relies on mathematical modeling that can describe the underlying dynamic behavior. In this work, we present a data-driven methodology for modeling the dynamics of nonlinear systems. To simplify this task, we aim to identify a coordinate transformation that allows us to represent the dynamics of nonlinear systems using a common, simple model structure. The advantage of a common simple model is that customized design tools developed for it can be applied to study a large variety of nonlinear systems. The simplest common model -- one can think of -- is linear, but linear systems often fall short in accurately capturing the complex dynamics of nonlinear systems. In this work, we propose using quadratic systems as the common structure, inspired by the lifting principle. According to this principle, smooth nonlinear systems can be expressed as quadratic systems in suitable coordinates without approximation errors. However, finding these coordinates solely from data is challenging. Here, we leverage deep learning to identify such lifted coordinates using only data, enabling a quadratic dynamical system to describe the system's dynamics. Additionally, we discuss the asymptotic stability of these quadratic dynamical systems. We illustrate the approach using data collected from various numerical examples, demonstrating its superior performance with the existing well-known techniques.