Linearization is a powerful tool for assessing, understanding, and controlling complex nonlinear systems. Traditionally, such approaches have been limited to small regions of the phase space. However, recent advances—particularly those based on Koopman operator theory—have extended the validity of linearization techniques to larger domains. Despite this progress, their limitations for truly nonlinear systems, such as those with coexisting attractors, have been pointed out. This raises a fundamental question: Can truly nonlinear dynamics be linearized? In this talk, we answer this question affirmatively for prototypical nonlinear systems with i) a continuous spectrum, ii) limit cycles, and iii) coexisting solutions. We explicitly construct linear systems that replicate these nonlinear behaviors and employ deep learning to approximate a mapping between linear and nonlinear dynamics. This approach yields low-dimensional linearizations in closed form. We illustrate this approach through application to hallmark nonlinear systems: the pendulum, the Duffing oscillator, and the Van-der-Pol oscillator. Finally, we compare our method to existing linearization techniques and discuss its implications for studying bifurcations.
A Wed, study studied this question.