In this talk, we present a sequence of works on the use of machine learning to discover mathematical structure from data in nonlinear wave systems. These include the identification of governing equations, reduced-order closure models, conservation laws, Lax pairs, and integrability. We will present a range of machine learning approaches, highlighting both their capabilities and their limitations, along with strategies to address these challenges. The effectiveness of these methods will be demonstrated on several nonlinear wave systems, illustrating their potential for advancing data-driven scientific discovery and revealing hidden mathematical structure.