High-energy-density physics is the field of physics concerned with studying matter at extremely high temperatures and densities. Such conditions produce highly nonlinear plasmas, in which several phenomena that can normally be treated independently of one another become strongly coupled. The study of these plasmas is important for our understanding of astrophysics, nuclear fusion and fundamental physics-however, the nonlinearities and strong couplings present in these extreme physical systems makes them very difficult to understand theoretically or to optimize experimentally. Here we argue that machine learning models and data-driven methods are in the process of reshaping our exploration of these extreme systems that have hitherto proved far too nonlinear for human researchers. From a fundamental perspective, our understanding can be improved by the way in which machine learning models can rapidly discover complex interactions in large datasets. From a practical point of view, the newest generation of extreme physics facilities can perform experiments multiple times a second (as opposed to approximately daily), thus moving away from human-based control towards automatic control based on real-time interpretation of diagnostic data and updates of the physics model. To make the most of these emerging opportunities, we suggest proposals for the community in terms of research design, training, best practice and support for synthetic diagnostics and data analysis.