Abstract: Science and engineering have long relied on interpretable, parsimonious models of signals and systems. Even in today’s era of highly parameterized models and abundant data, efficient and reliable designs require an understanding of how information flows across space, time, and scales. I will argue that phase spaces—familiar from microlocal analysis and statistical mechanics—offer a powerful bridge between signal processing, machine learning, and the physical world, with the potential to realize this understanding. Drawing on joint work with students and collaborators, I will show how a phase-space perspective illuminates fundamental questions in sampling, approximation, and learning theory for physical systems, and how it enables a new generation of scalable algorithms for inverse problems and simulation.