Learning Global Linear Representations of Nonlinear Dynamics

TL;DR

This paper demonstrates that certain complex nonlinear dynamical systems can be globally linearized using deep neural networks to approximate transformations, extending the applicability of linear models to nonlinear dynamics with coexisting attractors.

Contribution

The authors explicitly construct linear systems for complex nonlinear behaviors and develop neural network-based transformations, enabling finite-dimensional linearizations of nonlinear systems.

Findings

Successfully linearized systems with coexisting attractors

Deep neural networks approximate transformations between nonlinear and linear systems

Finite-dimensional linearizations exceed phase space dimension by at most one

Abstract

While linear systems are well-understood, no explicit solution for general nonlinear systems exists. A classical approach to make the understanding of linear system available in the nonlinear setting is to represent a nonlinear system by a linear model. While progress has been made in extending linearization techniques to larger domains and more complex attractor geometries, recent work has highlighted the limitations of these techniques when applied to nonlinear dynamics, such as those with coexisting attractors. In this work, we show nonlinear dynamics with a continuous Koopman spectrum, a limit cycle, and coexisting solutions that can be globally linearized. To this end, we explicitly construct linear systems mimicking these nonlinear behaviors. Subsequently, we approximate transformations between linear and nonlinear systems with deep neural networks. This approach yields finite dimensional linearizations exceeding the phase space dimension of the underlying linear system by one at most.