This paper addresses the challenge of identifying coordinate transformations that linearize strongly nonlinear dynamics, a critical step for applying standard linear theory to complex systems. The authors leverage deep learning to discover representations of Koopman eigenfunctions from trajectory data, aiming to create parsimonious and interpretable models. The Koopman operator, which linearizes dynamics through its eigenfunctions, has been a leading data-driven embedding method, but identifying these eigenfunctions is mathematically and computationally challenging. The proposed approach uses a modified auto-encoder to identify nonlinear coordinates on which the dynamics are globally linear. It also generalizes Koopman representations to systems with continuous spectra, such as the simple pendulum and nonlinear optics, by parameterizing the continuous frequency using an auxiliary network. This framework ensures compact and efficient embeddings while maintaining physical interpretability. The method is validated on several examples, including a simple model with a discrete spectrum, the nonlinear pendulum, and high-dimensional fluid flow past a cylinder, demonstrating its effectiveness in capturing complex dynamics with interpretable Koopman eigenfunctions.This paper addresses the challenge of identifying coordinate transformations that linearize strongly nonlinear dynamics, a critical step for applying standard linear theory to complex systems. The authors leverage deep learning to discover representations of Koopman eigenfunctions from trajectory data, aiming to create parsimonious and interpretable models. The Koopman operator, which linearizes dynamics through its eigenfunctions, has been a leading data-driven embedding method, but identifying these eigenfunctions is mathematically and computationally challenging. The proposed approach uses a modified auto-encoder to identify nonlinear coordinates on which the dynamics are globally linear. It also generalizes Koopman representations to systems with continuous spectra, such as the simple pendulum and nonlinear optics, by parameterizing the continuous frequency using an auxiliary network. This framework ensures compact and efficient embeddings while maintaining physical interpretability. The method is validated on several examples, including a simple model with a discrete spectrum, the nonlinear pendulum, and high-dimensional fluid flow past a cylinder, demonstrating its effectiveness in capturing complex dynamics with interpretable Koopman eigenfunctions.