Koopman operators provide a linear operator approach to nonlinear dynamics via their spectral properties. Extracting this information from trajectory data has become a significant focus, yet standard discretisation-based methods often fail. Chapter 11 builds a rigorous, practitioner-ready framework for data-driven Koopman spectral analysis, with case studies spanning chaotic and Hamiltonian dynamics, climate systems, shockwaves, and turbulent flows. After a focused primer, we analyse DMD and EDMD and present concrete failures on benchmark systems such as the Lorenz and Duffing systems. We then introduce Residual DMD (ResDMD), a data-driven analogue of operator folding that uses residuals to certify eigenpairs and reject spurious eigenvalues; kernelised variants extend the approach to high-dimensional settings via RKHS observables and a dual viewpoint through the adjoint Koopman operator. For measure-preserving dynamics, we develop mpEDMD and smoothing schemes based on correlations and resolvents to compute spectral measures and spectral type. Finally, adversarial examples and SCI lower bounds expose intrinsic limits of Koopman learning, including for random trajectory sampling.

Distinguishing between different phases of matter and detecting phase transitions are some of the most central tasks in many-body physics. Traditionally, these tasks are accomplished by searching for a small set of low-dimensional quantities capturing the macroscopic properties of each phase of the system, so-called order parameters. Because of the large state space underlying many-body systems, success generally requires a great deal of human intuition and understanding. In particular, it can be challenging to define an appropriate order parameter if the symmetry breaking pattern is unknown or the phase is of topological nature and thus exhibits nonlocal order. In this chapter, we explore the use of machine learning to automate the task of classifying phases of matter and detecting phase transitions. We discuss the application of various machine learning techniques, ranging from clustering to supervised learning and anomaly detection, to different physical systems, including the prototypical Ising model that features a symmetry-breaking phase transition and the Ising gauge theory which hosts a topological phase of matter.

This chapter highlights the use of data-driven discovery to address remaining gaps in our understanding of deep carbon.