This paper presents a deep learning approach for discovering and representing Koopman eigenfunctions of nonlinear dynamical systems. The Koopman operator provides a linear framework for analyzing nonlinear systems by identifying intrinsic coordinates that globally linearize the dynamics. However, obtaining these eigenfunctions has been mathematically and computationally challenging. The authors propose a deep learning framework that leverages the power of neural networks to discover these eigenfunctions from trajectory data. The proposed method uses a deep autoencoder to identify intrinsic coordinates that linearize the dynamics, while also incorporating an auxiliary network to parameterize the continuous frequency spectrum. This approach enables a compact and efficient representation of the dynamics, while maintaining the physical interpretability of Koopman embeddings. The framework is tested on several example systems, including a simple model with a discrete spectrum, a nonlinear pendulum with a continuous spectrum, and a high-dimensional fluid flow problem. The deep learning approach is trained using three types of loss functions: reconstruction loss, linear dynamics loss, and future state prediction loss. The network is designed to be parsimonious and interpretable, with the ability to handle both discrete and continuous spectra. The method is shown to accurately identify Koopman eigenfunctions, providing a globally linear representation of the dynamics in intrinsic coordinates. The results demonstrate that the deep learning approach can effectively identify Koopman eigenfunctions for a wide range of nonlinear systems, including those with continuous spectra. The framework is particularly effective for systems where traditional Koopman methods struggle due to the complexity of the continuous spectrum. The method is also shown to be efficient and interpretable, making it a promising tool for the analysis and control of nonlinear systems.This paper presents a deep learning approach for discovering and representing Koopman eigenfunctions of nonlinear dynamical systems. The Koopman operator provides a linear framework for analyzing nonlinear systems by identifying intrinsic coordinates that globally linearize the dynamics. However, obtaining these eigenfunctions has been mathematically and computationally challenging. The authors propose a deep learning framework that leverages the power of neural networks to discover these eigenfunctions from trajectory data. The proposed method uses a deep autoencoder to identify intrinsic coordinates that linearize the dynamics, while also incorporating an auxiliary network to parameterize the continuous frequency spectrum. This approach enables a compact and efficient representation of the dynamics, while maintaining the physical interpretability of Koopman embeddings. The framework is tested on several example systems, including a simple model with a discrete spectrum, a nonlinear pendulum with a continuous spectrum, and a high-dimensional fluid flow problem. The deep learning approach is trained using three types of loss functions: reconstruction loss, linear dynamics loss, and future state prediction loss. The network is designed to be parsimonious and interpretable, with the ability to handle both discrete and continuous spectra. The method is shown to accurately identify Koopman eigenfunctions, providing a globally linear representation of the dynamics in intrinsic coordinates. The results demonstrate that the deep learning approach can effectively identify Koopman eigenfunctions for a wide range of nonlinear systems, including those with continuous spectra. The framework is particularly effective for systems where traditional Koopman methods struggle due to the complexity of the continuous spectrum. The method is also shown to be efficient and interpretable, making it a promising tool for the analysis and control of nonlinear systems.